000$2020$ Mathematics Subject Classification. 11D09, 11R18, 11T22,
11T24.000Keywords and phrases. Jacobi sums, Gauss sums, Gaussian periods,
multiplication matrices, finite Fourier transform.000This work was partially
supported by JSPS KAKENHI Grant Number 19K03418.

# Davenport and Hasse’s theorems and lifts of multiplication matrices of
Gaussian periods

Akinari Hoshi  Department of Mathematics, Niigata University, Niigata
950-2181, Japan<EMAIL_ADDRESS>and  Kazuki Kanai  Graduate
School of Science and Technology, Niigata University, Niigata 950-2181, Japan
<EMAIL_ADDRESS>
###### Abstract.

Let $e\geq 2$ be an integer, $p^{r}$ be a prime power with $p^{r}\equiv 1\
({\rm mod}\ e)$ and $\eta_{r}(i)$ be Gaussian periods of degree $e$ for
$\mathbb{F}_{p^{r}}$. By the dual form of Davenport and Hasse’s lifting
theorem on Gauss sums, we establish lifts of the multiplication matrices of
the Gaussian periods $\eta_{r}(0),\ldots,\eta_{r}(e-1)$ which are defined by
F. Thaine. We also give some examples of the explicit lifts for prime degree
$e$ with $3\leq e\leq 23$ which also illustrate relations among lifts of
Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.

###### Contents

  1. 1 Introduction
  2. 2 Thaine’s results: compositions of multiplication matrices of Gaussian periods
    1. 2.1 Multiplication matrices of roots
    2. 2.2 The $d$-compositions of multiplication matrices
    3. 2.3 Multiplication matrices of Gaussian periods
  3. 3 The $d$-compositions
    1. 3.1 The $d$-compositions of matrices
    2. 3.2 The $d$-compositions of functions
  4. 4 Proof of Theorem 1.3
  5. 5 Examples of Theorem 1.3
  6. 6 Proof of Theorem 1.4
  7. 7 Applications of Theorem 1.4

##  1\. Introduction

Let $e\geq 2$ be an integer and $p^{r}$ be a prime power with $p^{r}\equiv 1\
({\rm mod}\ e)$. Write $p^{r}=ef+1$. Let $\mathbb{F}_{p^{r}}$ be the finite
field of $p^{r}$ elements and $\gamma$ be a fixed generator of
$\mathbb{F}_{p^{r}}^{\times}=\mathbb{F}_{p^{r}}\setminus\\{0\\}$. Let
$\zeta_{n}=e^{2\pi i/n}$ be an $n$-th root of unity. For $0\leq i\leq e-1$,
Gaussian periods $\eta_{r}(i)$ of degree $e$ for $\mathbb{F}_{p^{r}}$ are
defined by

$\displaystyle\eta_{r}(i):=\sum^{f-1}_{j=0}\zeta_{p}^{\mathrm{Tr}(\gamma^{ej+i})}$

where $\mathrm{Tr}$ is the trace map
$\mathrm{Tr}:\mathbb{F}_{p^{r}}\to\mathbb{F}_{p}$, and the period polynomial
$P_{e,r}(X)$ of degree $e$ for $\mathbb{F}_{p^{r}}$ is defined by
$P_{e,r}(X)=\prod^{e-1}_{i=0}\left(X-\eta_{r}(i)\right)\in\mathbb{Z}[X]$. Note
that $\eta_{r}(i)$ depends on the choice of $\gamma$.

We recall notion of Jacobi sums, Gauss sums and cyclotomic numbers and their
relations (see Berndt, Evans and Williams [BEW98]). For a nontrivial character
$\psi$ on $\mathbb{F}_{p^{r}}^{\times}$ and the trivial character
$\varepsilon$ on $\mathbb{F}_{p^{r}}^{\times}$, we extend them to
$\mathbb{F}_{p^{r}}$ by setting $\psi(0)=0$ and $\varepsilon(0)=1$. Let
$\psi_{1}$, $\psi_{2}$ be characters on $\mathbb{F}_{p^{r}}$. Jacobi sums
$J_{r}(\psi_{1},\psi_{2})$ and $J_{r}^{\ast}(\psi_{1},\psi_{2})$ for
$\mathbb{F}_{p^{r}}$ are defined by

$J_{r}(\psi_{1},\psi_{2}):=\sum_{\alpha\in\mathbb{F}_{p^{r}}}\psi_{1}(\alpha)\psi_{2}(1-\alpha)\quad\mbox{and}\quad
J_{r}^{\ast}(\psi_{1},\psi_{2}):=\sum_{\begin{subarray}{c}\alpha\in\mathbb{F}_{p^{r}}\\\
\alpha\neq 0,1\end{subarray}}\psi_{1}(\alpha)\psi_{2}(1-\alpha).$

If $\psi_{1},\psi_{2}\neq\varepsilon$, then
$J_{r}(\psi_{1},\psi_{2})=J_{r}^{\ast}(\psi_{1},\psi_{2})$ although
$J_{r}(\varepsilon,\psi_{2})=J_{r}(\psi_{1},\varepsilon)=0$,
$J_{r}(\varepsilon,\varepsilon)=p^{r}$ and
$J_{r}^{\ast}(\varepsilon,\psi_{2})=J_{r}^{\ast}(\psi_{1},\varepsilon)=-1$,
$J_{r}^{\ast}(\varepsilon,\varepsilon)=p^{r}-2$. Gauss sums $G_{r}(\psi_{1})$
and $G_{r}^{\ast}(\psi_{1})$ for $\mathbb{F}_{p^{r}}$ are defined to be

$G_{r}(\psi_{1}):=\sum_{\alpha\in\mathbb{F}_{p^{r}}}\psi_{1}(\alpha)\zeta_{p}^{{\rm
Tr}(\alpha)}\quad\mbox{and}\quad
G_{r}^{\ast}(\psi_{1}):=\sum_{\alpha\in\mathbb{F}_{p^{r}}^{\times}}\psi_{1}(\alpha)\zeta_{p}^{{\rm
Tr}(\alpha)}.$

If $\psi_{1}\neq\varepsilon$, then $G_{r}(\psi_{1})=G_{r}^{\ast}(\psi_{1})$
although $G_{r}(\varepsilon)=0$ and $G_{r}^{\ast}(\varepsilon)=-1$. We have
the well-known relations

$J_{r}(\psi_{1},\psi_{2})=\frac{G_{r}(\psi_{1})G_{r}(\psi_{2})}{G_{r}(\psi_{1}\psi_{2})}\quad\mbox{and}\quad
J_{r}^{\ast}(\psi_{1},\psi_{2})=\frac{G_{r}^{\ast}(\psi_{1})G_{r}^{\ast}(\psi_{2})}{G_{r}^{\ast}(\psi_{1}\psi_{2})}$

whenever $\psi_{1}\psi_{2}\neq\varepsilon$.

The following theorem is the Davenport and Hasse’s lifting theorem (see also
Weil [Wei49, pages 503–505], [BEW98, page 360, Theorem 11.5.2]):

######  Theorem 1.1 (Davenport and Hasse [DH35, Relation (0.8)]).

Let $e\geq 2$ be an integer, $p^{r}$ be a prime power and $\psi$ be a
nontrivial character on $\mathbb{F}_{p^{r}}$. Then, for any integer $n\geq 1$,
we have

$G_{nr}(\psi^{\prime})=(-1)^{n-1}G_{r}(\psi)^{n}$

where $\psi^{\prime}$ is the lift of $\psi$ from $\mathbb{F}_{p^{r}}$ to
$\mathbb{F}_{p^{nr}}$ defined by
$\psi^{\prime}(\alpha)=\psi(\mathrm{Nr}(\alpha))$ and $\mathrm{Nr}$ is the
norm map. In particular, if $\psi_{1}$, $\psi_{2}$ and $\psi_{1}\psi_{2}$ are
nontrivial characters on $\mathbb{F}_{p^{r}}$, then we have

$J_{nr}(\psi_{1}^{\prime},\psi_{2}^{\prime})=(-1)^{n-1}J_{r}(\psi_{1},\psi_{2})^{n}.$

From now on, we take the character $\chi$ of order $e$ on $\mathbb{F}_{p^{r}}$
with $\chi(\gamma)=\zeta_{e}$ and $\chi(0)=0$ where
$\langle\gamma\rangle=\mathbb{F}_{p^{r}}^{\times}$. For $0\leq i,j\leq e-1$,
we simply write

$J_{r}(i,j):=J_{r}(\chi^{i},\chi^{j})\quad\mbox{and}\quad
J_{r}^{\ast}(i,j):=J_{r}^{\ast}(\chi^{i},\chi^{j}).$

We have

$J_{r}(i,j)=J_{r}^{\ast}(i,j)+\delta_{i,0}+\delta_{0,j}$

where $\delta_{i,j}$ is Kronecker’s delta.

######  Remark 1.2.

(1) If we adopt the another manner $\varepsilon(0)=0$, then we have
$J_{r}(i,j)=J_{r}^{\ast}(i,j)$ (e.g. Myerson [Mye81], Parnami, Agrawal and
Rajwade [PAR82], Katre and Rajwade [KR85a]).
(2) We may take another Jacobi sum

$\displaystyle
J_{r}^{\prime}(i,j)=\sum_{\alpha\in\mathbb{F}_{p^{r}}}\chi^{i}(\alpha)\chi^{j}(1+\alpha)$

which satisfies
$J_{r}^{\prime}(i,j)=\chi(-1)^{i}J_{r}(i,j)=(-1)^{fi}J_{r}(i,j)$ (see e.g.
[PAR82], [KR85a]).

For $0\leq i,j\leq e-1$, cyclotomic numbers ${\rm Cyc}_{r}(i,j)$ of order $e$
for $\mathbb{F}_{p^{r}}$ are defined by

$\displaystyle{\rm Cyc}_{r}(i,j):=\\#\\{(v_{1},v_{2})\mid 0\leq
v_{1},v_{2}\leq f-1,\ 1+\gamma^{ev_{1}+i}\equiv\gamma^{ev_{2}+j}\ ({\rm mod}\
p^{r})\\}.$

We have the following well-known relations between cyclotomic numbers ${\rm
Cyc}_{r}(a,b)$ and Jacobi sums $J_{r}^{\ast}(i,j)$ (see [BEW98, page 79,
Theorem 2.5.1]):

$\sum_{i=0}^{e-1}\sum_{j=0}^{e-1}(-1)^{fi}\zeta_{e}^{-(ai+bj)}J_{r}^{\ast}(i,j)=e^{2}{\rm
Cyc}_{r}(a,b)$

and

$(-1)^{fi}\sum_{a=0}^{e-1}\sum_{b=0}^{e-1}{\rm
Cyc}_{r}(a,b)\zeta_{e}^{ai+bj}=J_{r}^{\ast}(i,j).$

Note that both ${\rm Cyc}_{r}(a,b)$ and $J_{r}^{\ast}(i,j)$ depend on the
choice of the fixed generator $\gamma$ of $\mathbb{F}_{p^{r}}^{\times}$.

We see that the product of the Gaussian periods is represented by a linear
combination of the Gaussian periods again and these coefficients are given in
terms of the cyclotomic numbers (see [BEW98, page 328, Lemma 10.10.2, page
437, Exercise 12.23]):

(1) $\displaystyle\eta_{r}(m)\eta_{r}(m+i)=\sum^{e-1}_{j=0}({\rm
Cyc}_{r}(i,j)-D_{i}f)\eta_{r}(m+j)$

where $D_{i}=\delta_{0,i}$ (resp. $\delta_{e/2,i}$) if $f$ is even (resp.
odd). It follows that the Gaussian periods are the eigenvalues of the $e\times
e$ matrix $C_{r}:=[{\rm Cyc}_{r}(i,j)-D_{i}f]_{0\leq i,j\leq e-1}$ called the
multiplication matrix of $\eta_{r}(0),\ldots,\eta_{r}(e-1)$ (see Section 2).
Hence the period polynomial $P_{e,r}(X)$ can be obtained as the characteristic
polynomial ${\rm Char}_{X}(C_{r})$ of the multiplication matrix $C_{r}$.

F. Thaine investigated various properties and characterizations of Gaussian
periods, cyclotomic numbers and Jacobi sums with applications to the
construction of cyclic polynomials in the series of the papers [Tha96],
[Tha99], [Tha00], [Tha01], [Tha04], [Tha08] (see also Lehmer [Leh88], Schoof
and Washington [SW88], Hashimoto and Hoshi [HH05a], [HH05b]).

According to Thaine [Tha04, Section 2], for two $e\times e$ matrices
$A=[a_{i,j}]_{0\leq i,j\leq e-1}$, $B=[b_{i,j}]_{0\leq i,j\leq e-1}$ and
$d\in(\mathbb{Z}/e\mathbb{Z})\setminus\\{0\\}$, we define the $d$-composition
$A\overset{d}{\ast}B$ of $A$ and $B$ as

$A\overset{d}{\ast}B:=\left[\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}a_{s,t}b_{ds+i,dt+j}\right]_{0\leq
i,j\leq e-1}.$

For the multiplication matrix $C_{1}$ (resp. $C^{\prime}_{1}$) of Gaussian
periods $\eta_{1}(0),\ldots,\eta_{1}(e-1)$ (resp.
$\eta^{\prime}_{1}(0),\ldots,\eta^{\prime}_{1}(e-1)$) of degree $e$ for
$\mathbb{F}_{p^{1}}$, the $d$-composition
$C_{1}\overset{d}{\ast}C^{\prime}_{1}$ gives the multiplication matrix of
$\theta_{d,0},\ldots,\theta_{d,e-1}$ where
$\theta_{d,i}=\sum_{s=0}^{e-1}\eta_{1}(s)\eta^{\prime}_{1}(ds+i)$. Hence we
obtain the cyclic polynomial ${\rm
Char}_{X}(C_{1}\overset{d}{\ast}C^{\prime}_{1})$ which gives an intermediate
cyclic field $\mathbb{Q}(\theta_{d,i})$ of degree $e$ in the composite
bicyclic field
$\mathbb{Q}(\eta_{1}(0),\eta^{\prime}_{1}(0))\subset\mathbb{Q}(\zeta_{p},\zeta_{q})$
with $p\neq q$ (see Section 2 and Section 3).

In Section 4, we regard the Gaussian periods $\eta_{r}(i)$ of degree $e$ for
$\mathbb{F}_{p^{r}}$ as the function
$\eta_{r}:\mathbb{Z}/e\mathbb{Z}\rightarrow\mathbb{C}$, $i\mapsto\eta_{r}(i)$
and the Gauss sums $G^{\ast}_{r}(\chi)$ for $\mathbb{F}_{p^{r}}$ as the
function $G^{\ast}_{r}:\widehat{\mathbb{Z}/e\mathbb{Z}}\rightarrow\mathbb{C}$,
$\chi\mapsto G^{\ast}_{r}(\chi)$. Then we find that they are each other’s
finite Fourier transform with some twist (see Lemma 4.1 in Section 4) and we
have:

######  Theorem 1.3 (Davenport and Hasse’s lifting theorem: the dual form).

Let $e\geq 2$ be an integer and $p^{r}$ be a prime power with $p^{r}\equiv 1\
({\rm mod}\ e)$. We regard the Gaussian periods $\eta_{r}(i)$ of degree $e$
for $\mathbb{F}_{p^{r}}$ as the function
$\eta_{r}:\mathbb{Z}/e\mathbb{Z}\rightarrow\mathbb{C}$, $i\mapsto\eta_{r}(i)$.
Then, for any integer $n\geq 1$, we have

$\eta_{nr}(i)=(-1)^{n-1}\eta_{r}^{(n)}(i)\quad\mbox{for}\quad 0\leq i\leq e-1$

where

$\displaystyle\eta_{r}^{(n)}(i)=\sum_{k_{1}+\cdots+k_{n}\equiv\,i\,({\rm
mod}\,e)\atop 0\leq k_{1},\ldots,k_{n}\leq
e-1}\eta_{r}(k_{1})\cdots\eta_{r}(k_{n})$

is the $n$-fold product of $\eta_{r}$ with respect to the convolution product.

By using Theorem 1.3, we get our main theorem which gives lifts of the
multiplication matrix $C_{r}$ of the Gaussian periods
$\eta_{r}(0),\ldots,\eta_{r}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{r}}$ via
Thaine’s $(-1)$-composition $\overset{-1}{\ast}$:

######  Theorem 1.4.

Let $e\geq 2$ be an integer and $p^{r}$ be a prime power with $p^{r}\equiv 1\
({\rm mod}\ e)$. Let $C_{r}=[{\rm Cyc}_{r}(i,j)-D_{i}f]_{0\leq i,j\leq e-1}$
be the multiplication matrix of the Gaussian periods
$\eta_{r}(0),\ldots,\eta_{r}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{r}}$.
Then, for any integer $n\geq 1$, we have

$C_{nr}=(-1)^{n-1}C_{r}^{(n)}$

where $C_{r}^{(n)}$ is the $n$-fold product of $C_{r}$ with respect to the
$(-1)$-composition $\overset{-1}{\ast}$. In particular, we have
$P_{e,nr}(X)={\rm Char}_{X}((-1)^{n-1}C_{r}^{(n)})$.

We organize this paper as follows. In Section 2, we review Thaine’s results on
the $d$-composition of the multiplication matrices based on [Tha04]. In
Section 3, we study the $d$-compositions of matrices and functions in the
general situations. This enables us to consider the $d$-compositions of the
multiplication matrices $C_{r}$ of the Gaussian periods and also the
$d$-compositions of the Gaussian periods $\eta_{r}(i)$ without linear
independence. In Section 4, we recall a Fourier transform on finite abelian
groups and show that Gaussian periods $\eta_{r}(i)$ and Gauss sums
$G_{r}^{*}(\chi)$ are each other’s finite Fourier transform (with some twist).
Using this, we give a proof of Theorem 1.3. We will give some examples of
Theorem 1.3 in Section 5. In Section 6, the proof of Theorem 1.4 will be
given. In Section 7, we give some examples of Theorem 1.4 for prime degree $e$
with $3\leq e\leq 23$ which also illustrate relations among lifts of Jacobi
sums, Gaussian periods and multiplication matrices of Gaussian periods as in
Theorem 1.1, Theorem 1.3 and Theorem 1.4 respectively.

##  2\. Thaine’s results: compositions of multiplication matrices of Gaussian
periods

We review Thaine’s results on the $d$-composition of the multiplication
matrices based on [Tha04, Section 1 and Section 2].

###  2.1. Multiplication matrices of roots

Thaine [Tha04, Section 1] defined the multiplication matrix $A$ of
$\theta_{0},\ldots,\theta_{e-1}$ as follows.

Let $D$ be an integrally closed domain with char $D=0$ and $K$ be the quotient
field of $D$. Let $e\geq 2$ be an integer and
$P(X)=\sum^{e}_{k=0}c_{k}X^{k}=\prod^{e-1}_{i=0}(X-\theta_{i})\in D[X]$ be a
cyclic polynomial, i.e. an irreducible polynomial with cyclic Galois group
over $K$. Then $K(\theta_{i})/K$ is a cyclic extension of degree $e$ with
$\mathrm{Gal}(K(\theta_{i})/K)=\langle\tau\rangle$. We may assume that
$\tau(\theta_{i})=\theta_{i+1}$ where we regard the subscripts modulo $e$. We
also assume that the discriminant
$\mathrm{disc}_{K(\theta_{i})/K}(\theta_{0},\ldots\theta_{e-1})=(\det[\tau^{j}(\theta_{i})]_{0\leq
i,j\leq e-1})^{2}\neq 0$. Then $\\{\theta_{0},\ldots,\theta_{e-1}\\}$ is a
vector space basis of $K(\theta_{i})$.

Let $M_{e}(K)$ be the algebra of $e\times e$ matrices over $K$. Define the
matrix $A=[a_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$ by
$\theta_{0}\theta_{i}=\sum^{e-1}_{j=0}a_{i,j}\theta_{j}$ which is called the
multiplication matrix of $\theta_{0},\ldots,\theta_{e-1}$. Note that the
$\theta_{i}$’s are eigenvalues of $A$ and hence $P(X)={\rm Char}_{X}(A)$; the
characteristic polynomial of $A$.

Thaine [Tha04, Proposition 1] showed that
(i) $a_{i,j}=a_{-i,j-i}$ for $0\leq i,j\leq e-1$;
(ii) $A(\mathcal{K}^{-i}A\mathcal{K}^{i})=(\mathcal{K}^{-i}A\mathcal{K}^{i})A$
for $0\leq i\leq e-1$ where $\mathcal{K}=[\delta_{i+1,j}]_{0\leq i,j\leq e-1}$
is the $e\times e$ circulant matrix

$\displaystyle\mathcal{K}=\left(\begin{array}[]{cccc}&1&&\\\ &&\ddots&\\\
&&&1\\\ 1&&&\end{array}\right).$

Conversely, if $A^{\prime}=[a^{\prime}_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$
satisfies (i) and (ii) and $P^{\prime}(X)={\rm Char}_{X}(A^{\prime})$ is
irreducible over $K$ with roots
$\theta^{\prime}_{0},\ldots,\theta^{\prime}_{e-1}$, then the Galois group of
$P^{\prime}(X)$ over $K$ is cyclic and $A^{\prime}$ is the multiplication
matrix of $\theta^{\prime}_{0},\ldots,\theta^{\prime}_{e-1}$ with ${\rm
disc}_{K(\theta_{i})/K}(\theta^{\prime}_{0},\ldots,\theta^{\prime}_{e-1})\neq
0$.

###  2.2. The $d$-compositions of multiplication matrices

Thaine [Tha04, Section 2] defined the $d$-composition
$A\overset{d}{\ast}A^{\prime}$ of the multiplication matrices $A$ and
$A^{\prime}$ as follows.

Let
$P(X)=\prod_{i=0}^{e-1}(X-\theta_{i}),P^{\prime}(X)=\prod_{i=0}^{e-1}(X-\theta^{\prime}_{i})\in
D[X]$ be cyclic polynomials of degree $e$. Then $L=K(\theta_{i})$ and
$L^{\prime}=K(\theta^{\prime}_{i})$ are cyclic extensions of $K$ of degree $e$
with $\mathrm{Gal}(L/K)=\langle\tau\rangle$, $\tau(\theta_{i})=\theta_{i+1}$
and $\mathrm{Gal}(L^{\prime}/K)=\langle\tau^{\prime}\rangle$,
$\tau^{\prime}(\theta^{\prime}_{i})=\theta^{\prime}_{i+1}$ where we regard the
subscripts modulo $e$.

We assume that $L\cap L^{\prime}=K$. We also assume that
$\\{\theta_{0},\ldots,\theta_{e-1}\\}$ and
$\\{\theta^{\prime}_{0},\ldots,\theta^{\prime}_{e-1}\\}$ are linearly
independent over $K$. Then $[LL^{\prime}:K]=e^{2}$ with $G={\rm
Gal}(LL^{\prime}/K)\simeq\langle\tau\rangle\times\langle\tau^{\prime}\rangle$
and $\\{\theta_{i}\theta^{\prime}_{j}\mid 0\leq i,j\leq e-1\\}$ becomes a
vector space basis of $LL^{\prime}$ over $K$. We make identifications
$\tau=(\tau,1)$ and $\tau^{\prime}=(1,\tau^{\prime})$. For $1\leq d\leq e-1$,
we have an intermediate field
$L_{d}=(LL^{\prime})^{\langle\tau{\tau^{\prime}}^{d}\rangle}=K(\theta_{d,i})$
which is a cyclic extension of $K$ of degree $e$ with a normal basis
$\\{\theta_{d,0},\ldots,\theta_{d,e-1}\\}$ where
$\theta_{d,i}=\sum^{e-1}_{s=0}\theta_{s}\theta^{\prime}_{ds+i}$.

Thaine [Tha04, Proposition 6] (see also [Tha04, Example 4]) proved that the
multiplication matrix $A_{d}$ of $\theta_{d,0},\ldots,\theta_{d,e-1}$ is given
by

$\displaystyle A_{d}=A\overset{d}{\ast}A^{\prime}$

where $A=[a_{i,j}]_{0\leq i,j\leq e-1}$ and
$A^{\prime}=[a^{\prime}_{i,j}]_{0\leq i,j\leq e-1}$ are the multiplication
matrices of $\theta_{0},\ldots,\theta_{e-1}$ and
$\theta^{\prime}_{0},\ldots,\theta^{\prime}_{e-1}$ respectively and the
$d$-composition $A\overset{d}{\ast}A^{\prime}$ of the matrices $A$ and
$A^{\prime}$ is defined by

$A\overset{d}{\ast}A^{\prime}:=\left[\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}a_{s,t}a^{\prime}_{ds+i,dt+j}\right]_{0\leq
i,j\leq e-1}.$

###  2.3. Multiplication matrices of Gaussian periods

The simplest example of the multiplication matrix is that of Gaussian periods
$\eta_{1}(0),\ldots,\eta_{1}(e-1)$ for $\mathbb{F}_{p^{1}}$ (see Section 1).

Let $p$ be a prime with $p\equiv 1\ ({\rm mod}\ e)$ and
$\eta_{1}(0),\ldots,\eta_{1}(e-1)$ be the Gaussian periods of degree $e$ for
$\mathbb{F}_{p^{1}}$. Then $\eta_{1}(0),\ldots,\eta_{1}(e-1)$ are linearly
independent over $\mathbb{Q}$ and $A=[{\rm Cyc}_{1}(i,j)-D_{i}f]_{0\leq
i,j\leq e-1}$ becomes the multiplication matrix of
$\eta_{1}(0),\ldots,\eta_{1}(e-1)$ with $P_{e,1}(X)={\rm Char}_{X}(A)$ (see
the equation (1) in Section 1).

However, in the general case with $r\geq 2$, Gaussian periods
$\eta_{r}(0),\ldots,\eta_{r}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{r}}$ are
not necessarily linearly independent over $\mathbb{Q}$. Myerson [Mye81] showed
that $P_{e,r}(X)$ splits over $\mathbb{Q}$ into
$\delta=\gcd(e,(p^{r}-1)/(p-1))$ factors. For example, $P_{e,e}(X)$ splits
completely over $\mathbb{Q}$, i.e.
$\eta_{e}(0),\ldots,\eta_{e}(e-1)\in\mathbb{Q}$.

In the next section, we study the $d$-compositions of matrices and the
$d$-compositions of functions for more general situations. This enables us to
consider the $d$-compositions of the multiplication matrices $C_{r}$ of the
Gaussian periods $\eta_{r}(0),\ldots,\eta_{r}(e-1)$ and also the
$d$-compositions of the Gaussian periods $\eta_{r}(i)$ without linear
independence.

##  3\. The $d$-compositions

###  3.1. The $d$-compositions of matrices

Let $K$ be a field with char $K=0$. Let $e\geq 2$ be an integer and $M_{e}(K)$
be the algebra of $e\times e$ matrices over $K$.

According to Thaine [Tha04] (see Subsection 2.2), for $A=[a_{i,j}]_{0\leq
i,j\leq e-1}$, $B=[b_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$ and
$d\in(\mathbb{Z}/e\mathbb{Z})\setminus\\{0\\}$, we define the $d$-composition
$A\overset{d}{\ast}B$ of $A$ and $B$ as

$A\overset{d}{\ast}B:=\left[\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}a_{s,t}b_{ds+i,dt+j}\right]_{0\leq
i,j\leq e-1}.$

We see that the $d$-composition $\overset{d}{\ast}$ and the ordinary addition
$+$ satisfy the distributive law, i.e.

$\displaystyle
A\overset{d}{\ast}(B+C)=(A\overset{d}{\ast}B)+(A\overset{d}{\ast}C),\
(A+B)\overset{d}{\ast}C=(A\overset{d}{\ast}C)+(B\overset{d}{\ast}C),$

although it does not satisfy the cancellation law, i.e. there exist matrices
$A,B,C$ such that $A\overset{d}{\ast}C=B\overset{d}{\ast}C$ and $A\neq B$.

######  Proposition 3.1.

Let $A=[a_{i,j}]_{0\leq i,j\leq e-1}$, $B=[b_{i,j}]_{0\leq i,j\leq e-1}\in
M_{e}(K)$. We write $A[i,j]=a_{i,j}$ for convenience. For
$d\in(\mathbb{Z}/e\mathbb{Z})^{\times}$, we have

$(B\overset{d}{\ast}A)[i,j]=(A\overset{d^{-1}}{\ast}B)[-d^{-1}i,-d^{-1}j].$

In particular, we get

$A\overset{-1}{\ast}B=B\overset{-1}{\ast}A.$

###### Proof.

Putting $s^{\prime}:=ds+i$ and $t^{\prime}:=dt+j$, we have

$\displaystyle(B\overset{d}{\ast}A)[i,j]$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}B[s,t]A[ds+i,dt+j]$
$\displaystyle=\sum^{e-1}_{s^{\prime}=0}\sum^{e-1}_{t^{\prime}=0}B[d^{-1}(s^{\prime}-i),d^{-1}(t^{\prime}-j)]A[s^{\prime},t^{\prime}]$
$\displaystyle=\sum^{e-1}_{s^{\prime}=0}\sum^{e-1}_{t^{\prime}=0}A[s^{\prime},t^{\prime}]B[d^{-1}s^{\prime}+(-d^{-1}i),d^{-1}t^{\prime}+(-d^{-1}j)]$
$\displaystyle=(A\overset{d^{-1}}{\ast}B)[-d^{-1}i,-d^{-1}j].$

∎

######  Corollary 3.2.

We have ${\rm Char}_{X}(B\overset{d}{\ast}A)={\rm
Char}_{X}(A\overset{d^{-1}}{\ast}B)$ where ${\rm Char}_{X}(A)$ stands for the
characteristic polynomial of $A$.

###### Proof.

By Proposition 3.1, we see that there exists an invertible matrix $P\in
M_{e}(K)$ such that

$B\overset{d}{\ast}A=P^{-1}(A\overset{d^{-1}}{\ast}B)P.$

Hence the assertion follows. ∎

######  Lemma 3.3.

Let $A=[a_{i,j}]_{0\leq i,j\leq e-1}$, $B=[b_{i,j}]_{0\leq i,j\leq e-1}\in
M_{e}(K)$. Let $\mathcal{K}=[\delta_{i+1,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$
be a circulant matrix. For $0\leq l\leq e-1$, we have
(i)
$\mathcal{K}^{l}(A\overset{d}{\ast}B)=A\overset{d}{\ast}(\mathcal{K}^{l}B)=(\mathcal{K}^{-d^{-1}l}A)\overset{d}{\ast}B$,
(ii)
$(A\overset{d}{\ast}B)\mathcal{K}^{l}=A\overset{d}{\ast}(B\mathcal{K}^{l})=(A\mathcal{K}^{-d^{-1}l})\overset{d}{\ast}B$.

###### Proof.

Write $A[i,j]=a_{i,j}$. We first see that

$(\mathcal{K}^{l}A)[i,j]=A[i+l,j],\ (A\mathcal{K}^{l})[i,j]=A[i,j-l].$

(i) The first equality follows from

$\displaystyle(\mathcal{K}^{l}(A\overset{d}{\ast}B))[i,j]$
$\displaystyle=(A\overset{d}{\ast}B)[i+l,j]$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]B[ds+(i+l),dt+j]$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t](\mathcal{K}^{l}B)[ds+i,dt+j]$
$\displaystyle=(A\overset{d}{\ast}(\mathcal{K}^{l}B))[i,j].$

By substituting $s^{\prime}:=s+d^{-1}l$, the second equality follows from

$\displaystyle(A\overset{d}{\ast}(\mathcal{K}^{l}B))[i,j]$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\mathcal{K}^{l}B[ds+i,dt+j]$
$\displaystyle=\sum^{e-1}_{s^{\prime}=0}\sum^{e-1}_{t=0}A[s^{\prime}-d^{-1}l,t]B[ds^{\prime}+i,dt+j]$
$\displaystyle=\sum^{e-1}_{s^{\prime}=0}\sum^{e-1}_{t=0}(\mathcal{K}^{-d^{-1}l})A[s^{\prime},t]B[ds^{\prime}+i,dt+j]$
$\displaystyle=((\mathcal{K}^{-d^{-1}l}A)\overset{d}{\ast}B)[i,j].$

(ii) can be proved in the similar way, and we omit the proof. ∎

######  Proposition 3.4.

For $A,B,C\in M_{e}(K)$ and $d_{1},d_{2}\in(\mathbb{Z}/e\mathbb{Z})^{\times}$,
we have

$A\overset{d_{1}}{\ast}(B\overset{d_{2}}{\ast}C)=(A\overset{-d_{2}^{-1}d_{1}}{\ast}B)\overset{d_{2}}{\ast}C.$

In particular, we get

$A\overset{d_{1}}{\ast}(B\overset{-1}{\ast}C)=(A\overset{d_{1}}{\ast}B)\overset{-1}{\ast}C$

and hence the $(-1)$-composition $\overset{-1}{\ast}$ satisfies the
associative law.

###### Proof.

For $A=[a_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$, we write $A[i,j]=a_{i,j}$.
We obtain the following expression of $A\overset{d}{\ast}B$ by using the
circulant matrix $\mathcal{K}=[\delta_{i+1,j}]_{0\leq i,j\leq e-1}\in
M_{e}(K)$:

$A\overset{d}{\ast}B=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\,\mathcal{K}^{ds}B\mathcal{K}^{-dt}.$

Then it follows from Lemma 3.3 that

$\displaystyle A\overset{d_{1}}{\ast}(B\overset{d_{2}}{\ast}C)$
$\displaystyle=-\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\,\mathcal{K}^{d_{1}s}(B\overset{d_{2}}{\ast}C)\mathcal{K}^{-d_{1}t}$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\left((\mathcal{K}^{-d_{2}^{-1}d_{1}s}B\mathcal{K}^{d_{2}^{-1}d_{1}t})\overset{d_{2}}{\ast}C\right)$
$\displaystyle=\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\left(\sum^{e-1}_{u=0}\sum^{e-1}_{v=0}\mathcal{K}^{-d_{2}^{-1}d_{1}s}B\mathcal{K}^{d_{2}^{-1}d_{1}t}[u,v]\,\mathcal{K}^{d_{2}u}C\mathcal{K}^{d_{2}v}\right)$
$\displaystyle=\sum^{e-1}_{u=0}\sum^{e-1}_{v=0}\left(\sum^{e-1}_{s=0}\sum^{e-1}_{t=0}A[s,t]\mathcal{K}^{-d_{2}^{-1}d_{1}s}B\mathcal{K}^{d_{2}^{-1}d_{1}t}\right)\\![u,v]\,\mathcal{K}^{d_{2}u}C\mathcal{K}^{d_{2}v}$
$\displaystyle=\sum^{e-1}_{u=0}\sum^{e-1}_{v=0}(A\overset{-d_{2}^{-1}d_{1}}{\ast}B)[u,v]\,\mathcal{K}^{d_{2}u}C\mathcal{K}^{d_{2}v}$
$\displaystyle=(A\overset{-d_{2}^{-1}d_{1}}{\ast}B)\overset{d_{2}}{\ast}C.$

The last assertion follows if we take $d_{2}=-1$. ∎

By the last statement of Proposition 3.4, we can define the $n$-fold product
$A^{(n)}$ as

$\displaystyle
A^{(n)}:=A\overset{-1}{\ast}\cdots\overset{-1}{\ast}A\quad(\text{$n$-fold}).$

###  3.2. The $d$-compositions of functions

Let $e\geq 2$ be an integer and
$P(X)=\sum^{e-1}_{k=0}c_{k}X^{k}=\prod^{e-1}_{i=0}(X-\theta_{i})\in D[X]$ be a
cyclic polynomial. Let $L^{2}(\mathbb{Z}/e\mathbb{Z})$ be the vector space of
all $\mathbb{C}$-valued functions on $\mathbb{Z}/e\mathbb{Z}$. We may regard
$\theta_{i}=\theta(i)$ as the function
$\theta:\mathbb{Z}/e\mathbb{Z}\to\mathbb{C}$, i.e. $\theta\in
L^{2}(\mathbb{Z}/e\mathbb{Z})$. Based on the results in Subsection 3.1, for
$f,g\in L^{2}(\mathbb{Z}/e\mathbb{Z})$ and
$d\in(\mathbb{Z}/e\mathbb{Z})\setminus\\{0\\}$, we define the $d$-composition
$f\overset{d}{\ast}g$ of $f$ and $g$ by

$(f\overset{d}{\ast}g)(i):=\sum^{e-1}_{s=0}f(s)g(ds+i).$

In particular, we get

$(f\overset{-1}{\ast}g)(i)=\sum_{k_{1}+k_{2}\equiv\,i\,({\rm mod}e)\atop 0\leq
k_{1},k_{2}\leq e-1}f(k_{1})g(k_{2})=(f*g)(i)$

where $*$ is the (usual) convolution of $L^{2}(\mathbb{Z}/e\mathbb{Z})$ (see
also Section 4). Hence the $(-1)$-composition $\overset{-1}{\ast}$ satisfies
the commutative and the associative laws, and we can define the $n$-fold
product of $f$ with respect to $\overset{-1}{\ast}$ as

$f^{\left(n\right)}(i):=(f\overset{-1}{\ast}\cdots\overset{-1}{\ast}f)(i)\quad(\text{$n$-fold}).$

By the definition, we get

$f^{(n)}(i)=\sum_{k_{1}+\cdots+k_{n}\equiv\,i\,({\rm mod}e)\atop 0\leq
k_{1},\ldots,k_{n}\leq e-1}f(k_{1})\cdots f(k_{n}).$

In order to prove Theorem 1.4, we need the following proposition which gives
the relation between $A\overset{d}{\ast}B$ and $f\overset{d}{\ast}g$:

######  Proposition 3.5.

Let $K$ be a field with ${\rm char}$ $K=0$. For $f,g\in
L^{2}(\mathbb{Z}/e\mathbb{Z})$, we assume that there exist $A=[a_{i,j}]_{0\leq
i,j\leq e-1}$, $B=[b_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$ such that

$f(i)\,f(j)=\sum^{e-1}_{k=0}a_{j-i,k-i}\,f(k)\ {\rm and}\
g(i)\,g(j)=\sum^{e-1}_{k=0}b_{j-i,k-i}\,g(k).$

Then we have

$(f\overset{d}{\ast}g)(i)\,(f\overset{d}{\ast}g)(j)=\sum^{e-1}_{k=0}(A\overset{d}{\ast}B)[j-i,k-i]\,(f\overset{d}{\ast}g)(k)$

where we write $A[i,j]=a_{i,j}$. In particular,

$f^{(n)}(i)\,f^{(n)}(j)=\sum^{e-1}_{j=0}A^{(n)}[j-i,k-i]\,f^{(n)}(j)$

where $A^{(n)}$ is the $n$-fold product of $A$ with respect to the
$(-1)$-composition $\overset{-1}{\ast}$.

###### Proof.

By substituting $k:=v-du$, $s:=n-m$, $t:=u-m$, we have

$\displaystyle(f\overset{d}{\ast}g)(i)(f\overset{d}{\ast}g)(j)$
$\displaystyle=\left(\sum_{m=0}^{e-1}f(m)g(dm+i)\right)\left(\sum_{n=0}^{e-1}f(n)g(dn+j)\right)$
$\displaystyle=\sum_{m,n=0}^{e-1}(f(m)f(n))(g(dm+i)g(dn+j))$
$\displaystyle=\sum_{m,n=0}^{e-1}\left(\sum_{u=0}^{e-1}A[n-m,u-m]f(u)\right)\left(\sum_{v=0}^{e-1}B[d(n-m)+j-i,v-(dm+i)]g(v)\right)$
$\displaystyle=\sum_{m,n,u,k=0}^{e-1}A[n-m,u-m]B[d(n-m)+j-i,d(u-n)+k-i]f(u)g(du+k)$
$\displaystyle=\sum_{s,t,u,k=0}^{e-1}A[s,t]B[ds+j-i,dt+k-i]f(u)g(du+k)$
$\displaystyle=\sum_{k=0}^{e-1}\left(\sum_{s,t=0}^{e-1}A[s,t]B[ds+(j-i),dt+(k-i)]\right)\left(\sum_{u=0}^{e-1}f(u)g(du+k)\right)$
$\displaystyle=\sum_{k=0}^{e-1}(A\overset{d}{\ast}B)[j-i,k-i](f\overset{d}{\ast}g)(k).$

∎

Applying Proposition 3.5 for Gaussian periods $\eta_{r}(i)$ of degree $e$ for
$\mathbb{F}_{p^{r}}$ and $\eta^{\prime}_{s}(i)$ for $\mathbb{F}_{q^{s}}$ with
$p^{r},q^{s}\equiv 1\ ({\rm mod}\ e)$ (we may apply the both cases $p\neq q$
and $p=q$) and their multiplication matrices, we get:

######  Corollary 3.6.

Let $e\geq 2$ be an integer and $p^{r}$ $($resp. $q^{s}$$)$ be a prime power
with $p^{r}\equiv 1\ ({\rm mod}\ e)$ $($resp. $q^{s}\equiv 1\ ({\rm mod}\
e)$$)$. $($We may take $p^{r}$, $q^{s}$ in the both cases $p\neq q$ and
$p=q$$.)$ We regard the Gaussian periods $\eta_{r}(i)$ of degree $e$ for
$\mathbb{F}_{p^{r}}$ as the functions from $\mathbb{Z}/e\mathbb{Z}$ to
$\mathbb{C}$, i.e. $\eta_{r}\in L^{2}(\mathbb{Z}/e\mathbb{Z})$. Let $C$
$($resp. $C^{\prime}$$)$ be the multiplication matrix of the Gaussian periods
$\eta_{r}(0),\ldots,\eta_{r}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{r}}$
$($resp. $\eta^{\prime}_{s}(0),\ldots,\eta^{\prime}_{s}(e-1)$ of degree $e$
for $\mathbb{F}_{q^{s}}$$)$. Then we have

$(\eta_{r}\overset{d}{\ast}\eta^{\prime}_{s})(i)\,(\eta_{r}\overset{d}{\ast}\eta^{\prime}_{s})(j)=\sum^{e-1}_{k=0}(C\overset{d}{\ast}C^{\prime})[j-i,k-i]\,(\eta_{r}\overset{d}{\ast}\eta^{\prime}_{s})(k).$

For the case $p\neq q$, we can find examples of $C\overset{d}{\ast}C^{\prime}$
in Thaine [Tha04, Example 4, page 259]. We will treat the case $p=q$ in the
remaining part of this paper.

##  4\. Proof of Theorem 1.3

We recall a Fourier transform on finite abelian groups (see Terras [Ter99,
Chapter 10]). Let $G$ be a finite abelian group and

$L^{2}(G)=\\{f:G\to\mathbb{C}\\}$

be the vector space of all $\mathbb{C}$-valued functions on $G$ with the inner
product $\langle f,g\rangle=\sum_{x\in G}f(x)\overline{g(x)}$. Let
$\widehat{G}={\rm Hom}(G,\mathbb{C}^{\times})$ be the dual of $G$. For $f\in
L^{2}(G)$, the finite Fourier transform $\mathscr{F}(f)=\widehat{f}\in
L^{2}(\widehat{G})$ of $f$ is defined to be

$\displaystyle(\mathscr{F}(f))(\chi)=\widehat{f}(\chi)=\sum_{x\in
G}f(x)\overline{\chi(x)}=\langle f,\chi\rangle.$

Then $\mathscr{F}:L^{2}(G)\rightarrow L^{2}(\widehat{G})$ becomes a bijective
linear transformation with the inverse

$\displaystyle(\mathscr{F}^{-1}(\widehat{f}))(x)=f(x)=\frac{1}{\\#G}\sum_{\chi\in\widehat{G}}\widehat{f}(\chi)\chi(x)=\frac{1}{\\#G}\sum_{\chi\in\widehat{G}}\langle
f,\chi\rangle\chi(x).$

For $f,g\in L^{2}(G)$, we define the convolution $f\ast g\in L^{2}(G)$ of $f$
and $g$ by

$(f\ast g)(x)=\sum_{y\in G}f(y)g(x-y).$

Then the space $L^{2}(G)$ with the convolution $\ast$ is isomorphic to the
group ring $\mathbb{C}[G]$ (with the usual convolution product) as a
commutative $\mathbb{C}$-algebra by $L^{2}(G)\ni f\mapsto\sum_{x\in
G}f(x)x\in\mathbb{C}[G]$. We also have the compatibility of the convolution
$\ast$ and the finite Fourier transform $\mathscr{F}(f)=\widehat{f}$:

(2) $\displaystyle\widehat{(f\ast
g)}(\chi)=\widehat{f}(\chi)\widehat{g}(\chi)$

(see Terras [Ter99, page 168, Theorem 2]).

In order to show Theorem 1.3, we prepare the following fundamental lemma:

######  Lemma 4.1.

Let $e\geq 2$ be an integer and $p^{r}$ be a prime power with $p^{r}\equiv 1\
({\rm mod}\ e)$. Let $\gamma$ be a fixed generator of
$\mathbb{F}_{p^{r}}^{\times}$ and $\chi$ be the character on
$\mathbb{F}_{p^{r}}$ with $\chi(\gamma)=\zeta_{e}$ and $\chi(0)=0$. We regard
the Gaussian periods $\eta_{r}(i)$ of degree $e$ for $\mathbb{F}_{p^{r}}$ as
the functions from $\mathbb{Z}/e\mathbb{Z}$ to $\mathbb{C}$, i.e. $\eta_{r}\in
L^{2}(\mathbb{Z}/e\mathbb{Z})$, and the Gauss sum $G_{r}^{\ast}(\chi^{j})$ for
$\mathbb{F}_{p^{r}}$ as the functions from $\widehat{\mathbb{Z}/e\mathbb{Z}}$
to $\mathbb{C}$, i.e. $G_{r}^{\ast}\in
L^{2}(\widehat{\mathbb{Z}/e\mathbb{Z}})$. Then the finite Fourier transform
$\mathscr{F}(\eta_{r})$ of $\eta_{r}$ is given by

$\displaystyle(\mathscr{F}(\eta_{r}))(\chi^{j})=G_{r}^{\ast}(\chi^{-j})$

and we also have

$\displaystyle(\mathscr{F}^{-1}(G_{r}^{\ast}))(i)=\eta_{r}(-i).$

###### Proof.

Because $\mathbb{F}_{p^{r}}^{\times}$ is a cyclic group of order $p^{r}-1=ef$
and $\chi\in\widehat{\mathbb{F}_{p^{r}}^{\times}}$ is of order $e$, the Gauss
sum $G_{r}^{\ast}(\chi^{j})$ can be regarded as the function from
$\widehat{\mathbb{Z}/e\mathbb{Z}}$ to $\mathbb{C}$ via factors through
$G_{r}^{\ast}:\widehat{\mathbb{F}_{p^{r}}^{\times}}\rightarrow\widehat{\mathbb{F}_{p^{r}}^{\times}}/H\rightarrow\mathbb{C}$
where $H$ is the group of $e$-th powers of
$\widehat{\mathbb{F}_{p^{r}}^{\times}}$ with
$\widehat{\mathbb{F}_{p^{r}}^{\times}}/H\simeq\widehat{\mathbb{Z}/e\mathbb{Z}}$.
Note that $\\{\chi^{j}H\mid 0\leq j\leq e-1\\}$ gives a set of complete
representatives for
$\widehat{\mathbb{F}_{p^{r}}^{\times}}/H\simeq\widehat{\mathbb{Z}/e\mathbb{Z}}$.

We have the following well-known relations between the Gauss sums
$G_{r}^{\ast}(\chi^{j})$ and the Gaussian periods $\eta_{r}(i)$:

$\displaystyle
G_{r}^{\ast}(\chi^{j})=\sum_{i=0}^{e-1}\zeta_{e}^{ij}\eta_{r}(i),\quad\eta_{r}(i)=\frac{1}{e}\sum_{j=0}^{e-1}\zeta_{e}^{-ij}G_{r}^{\ast}(\chi^{j})$

(see [Mye81, Proposition 1 (f)]). Then it follows that

$\displaystyle(\mathscr{F}(\eta_{r}))(\chi^{j})=\sum_{x\in\mathbb{Z}/e\mathbb{Z}}\eta_{r}(x)\overline{\chi^{j}}(x)=\sum_{i=0}^{e-1}\zeta_{e}^{-ij}\eta_{r}(i)=G_{r}^{\ast}(\chi^{-j})$

and

$\displaystyle(\mathscr{F}^{-1}(G_{r}^{\ast}))(i)$
$\displaystyle=\frac{1}{\\#(\mathbb{Z}/e\mathbb{Z})}\sum_{\psi\in\widehat{\mathbb{Z}/e\mathbb{Z}}}G_{r}^{\ast}(\psi)\psi(i)$
$\displaystyle=\frac{1}{e}\sum_{j=0}^{e-1}G_{r}^{\ast}(\chi^{j})\chi^{j}(i)=\frac{1}{e}\sum_{j=0}^{e-1}\zeta_{e}^{ij}G_{r}^{\ast}(\chi^{j})=\eta_{r}(-i).$

∎

Proof of Theorem 1.3. By Theorem 1.1, the equation (2) and Lemma 4.1, we have

$\displaystyle\widehat{\eta_{r}^{(n)}}(\chi^{i})=(\widehat{\eta_{r}}(\chi^{i}))^{n}=(G_{r}^{*}(\chi^{-i}))^{n}=(-1)^{n-1}G^{*}_{nr}(\chi^{-i})=(-1)^{n-1}\widehat{\eta_{nr}}(\chi^{i}).$

We get $\widehat{\eta_{r}^{(n)}}=(-1)^{n-1}\widehat{\eta_{nr}}$ and hence
$\eta_{r}^{(n)}=(-1)^{n-1}\eta_{nr}$.∎

##  5\. Examples of Theorem 1.3

Exponential Gauss sums $g_{r}(b,e)$ $(b\in\mathbb{F}_{p^{r}})$ of degree $e$
for $\mathbb{F}_{p^{r}}$ are defined by

$g_{r}(b,e):=\sum_{\alpha\in\mathbb{F}_{p^{r}}}\zeta_{p}^{{\rm
Tr}(b\alpha^{e})}.$

We write $g_{r}(e):=g_{r}(1,e)$. We also define the reduced Gaussian periods
$\eta_{r}^{*}(i)$ of order $e$ for $\mathbb{F}_{p^{r}}$ as

$\eta_{r}^{*}(i):=e\eta_{r}(i)+1=g_{r}(\gamma^{i},e)$

for $0\leq i\leq e-1$ (see [BEW98, page 327]). Let $\chi$ be the character on
$\mathbb{F}_{p^{r}}$ with $\chi(\gamma)=\zeta_{e}$ and $\chi(0)=0$ where
$\langle\gamma\rangle=\mathbb{F}_{p^{r}}^{\times}$ as before. We see that the
Gauss sums $G^{*}_{r}(\chi^{j})$ and the reduced Gaussian periods
$\eta_{r}^{*}(i)$ satisfy the following relations

$\displaystyle G^{*}_{r}(\chi^{j})$
$\displaystyle=\begin{cases}\displaystyle{\frac{1}{e}\sum_{i=0}^{e-1}\zeta_{e}^{ij}\eta^{*}_{r}(i)}&{\rm
if}\quad 1\leq j\leq e-1\\\ -1&{\rm if}\quad j=0,\end{cases}$
$\displaystyle\eta_{r}^{\ast}(i)$
$\displaystyle=\sum_{j=1}^{e-1}\zeta_{e}^{-ij}G^{*}_{r}(\chi^{j})$

(see [Mye81, Proposition 1 (g)], [BEW98, page 332, Theorem 10.10.8], cf. the
proof of Lemma 4.1). In particular, we have

$\displaystyle
g_{r}(e)=\eta^{*}_{r}(0)=\sum_{j=1}^{e-1}G^{*}_{r}(\chi^{j})=\sum_{j=1}^{e-1}G_{r}(\chi^{j}).$

We also use the reduced period polynomial
$P_{e,r}^{\ast}(X):=\prod^{e-1}_{i=0}\left(X-\eta^{\ast}_{r}(i)\right)$ of
degree $e$ for $\mathbb{F}_{p^{r}}$ with the coefficient of $X^{e-1}$ zero. An
explicit determination of the factors of $P_{e,r}^{*}(X)$ is important because
the exponential Gauss sum $g_{r}(e)=\eta^{*}_{r}(0)$ becomes a root of
$P_{e,r}^{*}(X)$ (see also Section 7).

By applying Theorem 1.3, we can obtain the reduced Gaussian periods
$\eta_{e}^{*}(i)$ $(i=0,\ldots,e-1)$ of order $e$ for $\mathbb{F}_{p^{e}}$ as
follows (we take the generator $\gamma$ of
$\mathbb{F}_{p}^{\times}=\langle\gamma\rangle$ as the smallest one):

(1) $e=3$. We take $p=7=ef+1$ with $f=2$ and $\gamma=3$. Then
$\eta_{3}^{*}(i)$ $(i=0,1,2)$ are given by $7$, $-35$, $28$ (in this order).

(2) $e=5$. We take $p=11=ef+1$ with $f=2$ and $\gamma=2$. Then
$\eta_{5}^{*}(i)$ $(i=0,\ldots,4)$ are given by $-979$, $-649$, $1276$, $-99$,
$451$.

(3) $e=7$. We take $p=29=ef+1$ with $f=4$ and $\gamma=2$. Then
$\eta_{7}^{*}(i)$ $(i=0,\ldots,6)$ are given by $-317869$, $-259405$,
$-324771$, $442569$, $233682$, $-182671$, $408465$.

(4) $e=11$. We take $p=23=ef+1$ with $f=2$ and $\gamma=5$. Then
$\eta_{11}^{*}(i)$ $(i=0,\ldots,10)$ are given by $52918009$, $3199967$,
$-202694722$, $-64390754$, $142959444$, $-23093817$, $166665038$, $-19592803$,
$47121273$, $-58652208$, $-44439427$.

(5) $e=13$. We take $p=53=ef+1$ with $f=4$ and $\gamma=2$. Then
$\eta_{13}^{*}(i)$ $(i=0,\ldots,12)$ are given by $782475795674$,
$338244988654$, $-245670171356$, $83828569254$, $-740552966334$,
$-910543059425$, $117899008800$, $664438112586$, $-186980700750$,
$-238169301889$, $-277653262665$, $1040615291340$, $-427932303889$.

(6) $e=17$. We take $p=103=ef+1$ with $f=6$ and $\gamma=5$. Then
$\eta_{17}^{*}(i)$ $(i=0,\ldots,16)$ are given by $-651513206543247755$,
$670088231006862759$, $-373934090375919493$,
$587253242462231659$, $-243310155546790559$, $163898849457734107$,
$-197783211402587952$,
$-1253189038565026183$, $35922811461007315$, $356621718684896633$,
$-478731856802195967$,
$-289516205265127375$, $461908111585063663$, $464742031061114921$,
$670357206530506901$,
$282238003107978403$, $-205052440856501077$.

We give GAP ([GAP]) computations for (6) $e=17$, $p=103$ and $\gamma=5$. The
cases (1)–(5) can be obtained by the similar manner.

    gap> etaf:=function(e,p,i)
    > local g;
    > g:=PrimitiveRootMod(p);
    > if i<0 then i:=i mod e;
    > fi;
    > return Sum([1..(p-1)/e],j->E(p)^(g^(e*j+i)));
    > end;
    function( e, p, i ) ... end
    gap> e:=17;;p:=103;;PrimitiveRootMod(p); # e=17, p=103, g=5
    5
    gap> eta:=function(i)
    > return(etaf(e,p,i));
    > end;
    function( i ) ... end
    gap> p2:=function(i)
    > return Sum([0..e-1],k1->eta(k1)*eta(-k1+i));
    > end;
    function( i ) ... end
    gap> p4:=function(i)
    > return Sum([0..e-1],k1->p2(k1)*p2(-k1+i));
    > end;
    function( i ) ... end
    gap> p8:=function(i)
    > return Sum([0..e-1],k1->p4(k1)*p4(-k1+i));
    > end;
    function( i ) ... end
    gap> p9:=function(i)
    > return Sum([0..e-1],k1->p8(k1)*eta(-k1+i));
    > end;
    function( i ) ... end
    gap> p17:=function(i)
    > return Sum([0..e-1],k1->p9(k1)*p8(-k1+i));
    > end;
    function( i ) ... end
    gap> L:=List([0..e-1],i->p17(i)); # Gaussian periods eta_{e,i}: i=0,...16
    [ -38324306267249868, 39416954765109574, -21996122963289382,
      34544308380131274, -14312362090987680, 9641108791631418,
      -11634306553093409, -73717002268530952, 2113106556529842,
      20977748157935096, -28160697458952704, -17030365015595728,
      27171065387356686, 27337766533006760, 39432776854735700,
      16602235476939906, -12061908285676534 ]
    gap> 17*L+1; # reduced Gaussian periods eta^*_{e,i}: i=0,...16
    [ -651513206543247755, 670088231006862759, -373934090375919493,
      587253242462231659, -243310155546790559, 163898849457734107,
      -197783211402587952, -1253189038565026183, 35922811461007315,
      356621718684896633, -478731856802195967, -289516205265127375,
      461908111585063663, 464742031061114921, 670357206530506901,
      282238003107978403, -205052440856501077 ]

##  6\. Proof of Theorem 1.4

For $A=[a_{i,j}]_{0\leq i,j\leq e-1}\in M_{e}(K)$, we write $A[i,j]=a_{i,j}$
for convenience. By the definition, for the multiplication matrix $C_{nr}$ of
Gaussian periods $\eta_{nr}(0),\ldots,\eta_{nr}(e-1)$, we have

$\eta_{nr}(i)\eta_{nr}(j)=\sum^{e-1}_{k=0}C_{nr}[j-i,k-i]\eta_{nr}(k).$

By Theorem 1.3 (the dual form of Davenport and Hasse’s lifting theorem), we
obtain

$\eta_{r}^{(n)}(i)\eta_{r}^{(n)}(j)=\sum^{e-1}_{k=0}(-1)^{n-1}C_{nr}[j-i,k-i]\eta_{r}^{(n)}(k).$

By setting $l:=j-i$ and $m:=k-i$, we get

(3)
$\displaystyle\eta_{r}^{(n)}(i)\eta_{r}^{(n)}(i+l)=\sum^{e-1}_{m=0}(-1)^{n-1}C_{nr}[l,m]\eta_{r}^{(n)}(i+m).$

On the other hand, by Proposition 3.5, we have

$\eta_{r}^{(n)}(i)\eta_{r}^{(n)}(j)=\sum^{e-1}_{k=0}C_{r}^{(n)}[j-i,k-i]\eta_{r}^{(n)}(k).$

By setting $l:=j-i$ and $m:=k-i$ also, we obtain

(4)
$\displaystyle\eta_{r}^{(n)}(i)\eta_{r}^{(n)}(i+l)=\sum^{e-1}_{m=0}C_{r}^{(n)}[l,m]\eta_{r}^{(n)}(i+m).$

Equations (3) and (4) imply that $(-1)^{n-1}C_{nr}$ and $C_{r}^{(n)}$ have the
same eigenvalues $\eta_{r}^{(n)}(i)$ $(0\leq i\leq e-1)$ and the same
eigenvector

$T_{i}=(\eta_{r}^{(n)}(i),\eta_{r}^{(n)}(i+1),\ldots,\eta_{r}^{(n)}(0),\eta_{r}^{(n)}(1),\ldots,\eta_{r}^{(n)}(i-1))^{t}$

with respect to $\eta_{r}^{(n)}(i)$ where $t$ stands for the transposed
vector.

We take the circulant matrix $P:=(T_{0},\ldots,T_{e-1})$ with determinant
$\prod_{j=0}^{e-1}f(\zeta_{e}^{j})$ where
$f(x)=\sum_{i=0}^{e-1}\eta_{r}^{(n)}(i)x^{i}$. We see that the matrix $P$ is
invertible because there exist at least two distinct
$\eta_{r}^{(n)}(i)=(-1)^{n-1}\eta_{nr}(i)$ by Baumert, Mills and Ward [BMW82,
Lemma 2 and the proof of Lemma 3]. Hence the both $(-1)^{n-1}C_{nr}$ and
$C_{r}^{(n)}$ are diagonalized by the same $P$:

$P^{-1}(-1)^{n-1}C_{nr}P=\begin{pmatrix}\eta_{r}^{(n)}(0)&&\\\ &\ddots&\\\
&&\eta_{r}^{(n)}(e-1)\\\ \end{pmatrix}=P^{-1}C_{r}^{(n)}P.$

This implies that $(-1)^{n-1}C_{nr}=C_{r}^{(n)}$. ∎

##  7\. Applications of Theorem 1.4

We give some applications of Theorem 1.4 when $e=l$ is an odd prime which
illustrate relations among lifts of Jacobi sums, Gaussian periods and
multiplication matrices of Gaussian periods as in Theorem 1.1, Theorem 1.3 and
Theorem 1.4 respectively. Note that Theorem 1.4 enables us to get lifts of
multiplication matrices of Gaussian periods within the base field $\mathbb{Q}$
although Davenport and Hasse’s lifting theorem for Jacobi sums needs to
consider the extended field $\mathbb{Q}(\zeta_{l})$. In particular, we can
recover results [Hos06, Theorem 1, Corollary 9] for the reduced period
polynomial $P_{5,5s}^{*}(X)$ and the exponential Gauss sums $g_{5s}(5)$ (see
(2) below).

Let $p$ be a prime with $p\equiv 1\ ({\rm mod}\ l)$. Katre and Rajwade [KR85a,
Main theorem, page 186] gave some system of Diophantine equations whose unique
solution gives the coefficient $a_{1}(n),\ldots,a_{l-1}(n)$ of the Jacobi sums
$J_{r}(1,n)=\sum_{i=1}^{l-1}a_{i}(n)\zeta_{l}^{i}\in\mathbb{Z}[\zeta_{l}]$
$(1\leq n\leq l-2)$ and the cyclotomic numbers ${\rm Cyc}_{r}(i,j)$ of order
$l$ are obtained in terms of $a_{1}(n),\ldots,a_{l-1}(n)$ $(1\leq n\leq l-2)$
as

$\displaystyle l^{2}{\rm Cyc}_{r}(0,0)$
$\displaystyle=p^{r}-3l+1-\sum_{n=1}^{l-2}\sum_{k=1}^{l-1}a_{k}(n),$
$\displaystyle l^{2}{\rm Cyc}_{r}(i,j)$
$\displaystyle=l\left(\delta_{i,0}+\delta_{0,j}+\delta_{i,j}+\sum_{n=1}^{l-2}a_{in+j}(n)\right)+l^{2}{\rm
Cyc}_{r}(0,0)$

where the subscripts in $a_{in+j}(n)$ are considered modulo $l$. See also van
Wamelen [Wam02] for general cases where $e\geq 3$ and $p^{r}\equiv 1\ ({\rm
mod}\ e)$. Recall that $C_{r}=[{\rm Cyc}_{r}(i,j)-D_{i}f]_{0\leq i,j\leq e-1}$
is the multiplication matrix of the Gaussian periods
$\eta_{r}(0),\ldots,\eta_{r}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{r}}$.
Hence Theorem 1.4 gives explicit lifts of not only the multiplication matrix
$C_{r}$ but also of cyclotomic numbers ${\rm Cyc}_{r}(i,j)$ and of Jacobi sums
$J_{r}(i,j)$ from $\mathbb{F}_{p^{r}}$ to $\mathbb{F}_{p^{nr}}$. Recall also
that the Jacobi sum
$J_{r}(i,j)=J_{r}(\chi^{i},\chi^{j})=\sum_{\alpha\in\mathbb{F}_{p^{r}}}\chi^{i}(\alpha)\chi^{j}(1-\alpha)$
where $\chi$ is a character of order $e$ on $\mathbb{F}_{p^{r}}$ with
$\chi(\gamma)=\zeta_{e}$, $\mathbb{F}_{p^{r}}^{\times}=\langle\gamma\rangle$
and $\chi(0)=0$.

Computing exponential Gauss sums $g_{er}(\lambda^{i},e)=\eta_{er}^{*}(i)$
($\mathbb{F}_{p^{er}}^{\times}=\langle\lambda\rangle$) is important because it
is equivalent to that of the weight distribution of irreducible cyclic codes
(see McEliece and Rumsey [MR72], Baumert and McEliece [BM72, Theorem 6],
McEliece [McE74, Section 2], Ding and Yang [DY13, Section 3] and [BEW98,
Section 11.7]).

We give applications of Theorem 1.4 for prime degree $e=l$ with $3\leq l\leq
23$.

(1) $e=3$ (cf. Gauss [Gau01, Section 358], Katre and Rajwade [KR85a,
Proposition 1], [BEW98, Section 3.1, Section 10.10]). Let $p$ be a prime with
$p\equiv 1\ ({\rm mod}\ 3)$. The Jacobi sum $J_{r}(1,1)$ is given by

$\displaystyle J_{r}(1,1)=J(c,d)=\frac{c+3d}{2}+3d\zeta_{3}$

where $c,d\in\mathbb{Z}$ are given as the integer solutions of the Diophantine
equation

(5) $\displaystyle\begin{cases}4p^{r}=c^{2}+27d^{2},\\\ c\equiv 1\ ({\rm mod}\
3),\ p\nmid c.\end{cases}$

The equations have two solutions $(c,\pm d)$ and the sign of $d$ depends on
the choice of $\gamma$. The unique solution with respect to $\gamma$ can be
determined by

$\displaystyle\gamma^{(p^{r}-1)/3}\equiv\frac{c+9d}{c-9d}\ ({\rm mod}\ p)$

(see Katre and Rajwade [KR85a, Proposition 1], [BEW98, Section 3.1]). The
multiplication matrix $C_{r}$ of the Gaussian periods
$\eta_{r}(0),\eta_{r}(1),\eta_{r}(2)$ of degree $3$ for $\mathbb{F}_{p^{r}}$
is given by

$\displaystyle C_{r}=C_{r}(p,c,d)=\left(\begin{array}[]{ccccc}A-f&B-f&C-f\\\
B&C&D\\\ C&D&B\\\ \end{array}\right)$

where

$\displaystyle A$ $\displaystyle={\rm Cyc}_{r}(0,0)=\frac{1}{9}(p^{r}+c-8),$
$\displaystyle B$ $\displaystyle={\rm
Cyc}_{r}(0,1)=\frac{1}{18}(2p^{r}-c+9d-4),$ $\displaystyle C$
$\displaystyle={\rm Cyc}_{r}(0,2)=\frac{1}{18}(2p^{r}-c-9d-4),$ $\displaystyle
D$ $\displaystyle={\rm Cyc}_{r}(1,2)=\frac{1}{9}(p^{r}+c+1)$

(see [BEW98, Section 2.3]). Then we have

$\displaystyle P_{3,r}(X)$ $\displaystyle={\rm Char}_{X}(C_{r}(p,c,d)),$
$\displaystyle P_{3,r}^{*}(X)$
$\displaystyle=P_{3,r}^{*}(p,c,d;X)=3^{3}P_{3,r}((X-1)/3)=X^{3}-3p^{r}X-p^{r}c.$

By Theorem 1.4, we get

$\displaystyle C_{nr}$
$\displaystyle=C_{nr}(p,c,d)=(-1)^{n-1}C_{r}(p,c,d)^{(n)}=C_{r}(p^{n},c^{(n)},d^{(n)})$

where $c^{(n)}$, $d^{(n)}$ can be obtained as a form of degree $n$ in $c$,
$d$. Note that $c^{(n)}$, $d^{(n)}$ satisfy the equation (5) with respect to
$p^{nr}$ instead of $p^{r}$. In particular, we get
$J_{nr}(1,1)=J(c^{(n)},d^{(n)})$ and

$\displaystyle
p^{nr}=\left(\frac{c^{2}+27d^{2}}{4}\right)^{n}=\frac{(c^{(n)})^{2}+27(d^{(n)})^{2}}{4}.$

For $n=2,3$, we can obtain that

$\displaystyle c^{(2)}$ $\displaystyle=\frac{1}{2}(-c^{2}+27d^{2}),\
d^{(2)}=-cd,$ $\displaystyle c^{(3)}$
$\displaystyle=\frac{1}{4}c(c+9d)(c-9d),\ d^{(3)}=\frac{3}{4}d(c+3d)(c-3d).$

and

$\displaystyle P_{3,2r}^{*}(X)=P_{3,r}^{*}(p^{2},c^{(2)},d^{(2)};X)$
$\displaystyle=X^{3}-3p^{2r}X-p^{2r}c^{(2)}$
$\displaystyle=X^{3}-3p^{2r}X-\frac{1}{2}p^{2r}(-c^{2}+27d^{2}),$
$\displaystyle P_{3,3r}^{*}(X)=P_{3,r}^{*}(p^{3},c^{(3)},d^{(3)};X)$
$\displaystyle=X^{3}-3p^{3r}X-p^{3r}c^{(3)}$
$\displaystyle=X^{3}-\frac{3}{4}p^{2r}(c^{2}+27d^{2})X-\frac{1}{4}\left(p^{3r}c(c+9d)(c-9d)\right)$
$\displaystyle=\left(X-p^{r}c\right)\left(X+p^{r}\,\frac{c+9d}{2}\right)\left(X+p^{r}\,\frac{c-9d}{2}\right)$

(see [BEW98, Section 12.10, page 427]). Because the exponential cubic Gauss
sum $g_{3r}(3)=\eta_{3r}^{*}(0)$ is one of the roots of $P_{3,3r}^{*}(X)$
which does not depend on the choice of $\gamma$, we get

$\displaystyle g_{3r}(3)=p^{r}c.$

Similarly, for $n=3m$, we have

$\displaystyle g_{3mr}(3)$ $\displaystyle=p^{mr}c^{(m)}.$

(2) $e=5$ (cf. Lehmer [Leh51, Equation (10)], Berndt and Evans [BE81, Section
5], Katre and Rajwade [KR85b], [BEW98, Section 3.7], Hoshi [Hos03, Section 5],
[Hos06, Section 3]). Let $p$ be a prime with $p\equiv 1\ ({\rm mod}\ 5)$. The
Jacobi sum $J_{r}(1,1)$ is given by

$\displaystyle
J_{r}(1,1)=J(x,w,v,u)=\frac{1}{4}\left(Z\zeta_{5}+\sigma^{3}(Z)\zeta_{5}^{2}+\sigma(Z)\zeta_{5}^{3}+\sigma^{2}(Z)\zeta_{5}^{4}\right)$

where $Z=-x+5w+4v+2u$, $\sigma(x,w,v,u)=(x,-w,-u,v)$ and
$x,w,v,u\in\mathbb{Z}$ are obtained as the integer solutions of the system of
Diophantine equations

(6) $\displaystyle\begin{cases}16p^{r}=x^{2}+125w^{2}+50v^{2}+50u^{2},\\\
xw=v^{2}-4vu-u^{2},\\\ x\equiv 1\ ({\rm mod}\ 5),\ p\nmid
x^{2}-125w^{2}.\end{cases}$

The equations have four solutions $\sigma^{i}(x,w,v,u)$ $(i=0,1,2,3)$ which
depend on the choice of $\gamma$. The unique solution with respect to $\gamma$
can be determined by

$\displaystyle\gamma^{(p^{r}-1)/5}\equiv\frac{x^{2}-125w^{2}-10(2xu-
xv-25wv)}{x^{2}-125w^{2}+10(2xu-xv-25wv)}\ ({\rm mod}\ p)$

(see Katre and Rajwade [KR85b, Theorem 1]). The multiplication matrix of
$C_{r}$ of the Gaussian periods $\eta_{r}(0),\ldots,\eta_{r}(4)$ of order $5$
for $\mathbb{F}_{p^{r}}$ is given by

$\displaystyle
C_{r}=C_{r}(p,x,u,v,w)=\left(\begin{array}[]{ccccc}A-f&B-f&C-f&D-f&E-f\\\
B&E&F&G&F\\\ C&F&D&G&G\\\ D&G&G&C&F\\\ E&F&G&F&B\\\ \end{array}\right)$

where

$\displaystyle A$ $\displaystyle=\frac{1}{25}(p^{r}+3x-14),$ $\displaystyle B$
$\displaystyle=\frac{1}{100}(4p^{r}-3x+25w+50v-16),$ $\displaystyle C$
$\displaystyle=\frac{1}{100}(4p^{r}-3x-25w+50u-16),$ $\displaystyle D$
$\displaystyle=\frac{1}{100}(4p^{r}-3x-25w-50u-16),$ $\displaystyle E$
$\displaystyle=\frac{1}{100}(4p^{r}-3x+25w-50v-16),$ $\displaystyle F$
$\displaystyle=\frac{1}{50}(2p^{r}+x-25w+2),$ $\displaystyle G$
$\displaystyle=\frac{1}{50}(2p^{r}+x+25w+2).$

Then we have

$\displaystyle P_{5,r}(X)$ $\displaystyle={\rm Char}_{X}(C_{r}(p,x,u,v,w)),$
$\displaystyle P_{5,r}^{*}(X)$
$\displaystyle=P_{5,r}^{*}(p,x,w,v,u;X)=5^{5}P_{5,r}((X-1)/5)$
$\displaystyle=X^{5}-10p^{r}X^{3}-5p^{r}xX^{2}+\frac{5}{4}p^{r}(4p^{r}-x^{2}+125w^{2})X+\frac{1}{8}p^{r}(-x^{3}+8p^{r}x+625w(v^{2}-u^{2}))$

(see [Leh51, Equation (10)], [BE81, Section 5], [Hos06, Section 3]).

By Theorem 1.4, we get

$\displaystyle
C_{nr}=C_{nr}(p,x,w,v,u)=(-1)^{n-1}C_{r}(p,x,w,v,u)^{(n)}=C_{r}(p^{n},x^{(n)},w^{(n)},v^{(n)},u^{(n)})$

where $x^{(n)}$, $w^{(n)}$, $v^{(n)}$, $u^{(n)}$ can be obtained as a form of
degree $n$ in $x$, $w$, $v$, $u$. Note that $x^{(n)}$, $w^{(n)}$, $v^{(n)}$,
$u^{(n)}$ satisfy the equation (6) with respect to $p^{nr}$ instead of
$p^{r}$. In particular, we have
$J_{nr}(1,1)=J(x^{(n)},w^{(n)},v^{(n)},u^{(n)})$,

$\displaystyle p^{nr}$
$\displaystyle=\left(\frac{x^{2}+125w^{2}+50v^{2}+50u^{2}}{16}\right)^{n}=\frac{(x^{(n)})^{2}+125(w^{(n)})^{2}+50(v^{(n)})^{2}+50(u^{(n)})^{2}}{16},$
$\displaystyle x^{(n)}w^{(n)}$
$\displaystyle=(v^{(n)})^{2}-4v^{(n)}u^{(n)}-(u^{(n)})^{2}.$

For $n=2$, we can obtain that

$\displaystyle x^{(2)}$
$\displaystyle=\frac{1}{4}(-x^{2}-125w^{2}+50v^{2}+50u^{2}),$ $\displaystyle
w^{(2)}$ $\displaystyle=\frac{1}{2}(-xw-v^{2}+4vu+u^{2}),$ $\displaystyle
v^{(2)}$ $\displaystyle=\frac{1}{2}(-xv-10wu+5vw),$ $\displaystyle u^{(2)}$
$\displaystyle=\frac{1}{2}(-xu-10wv-5uw).$

Continuing the argument, we get $x^{(5)}$, $w^{(5)}$, $v^{(5)}$, $u^{(5)}$ and
hence $P_{5,5r}(X)={\rm Char}_{X}(C_{r}^{(5)})$ and

$\displaystyle
P_{5,5r}^{*}(X)=\left(X-\frac{p^{r}}{16}L\right)\prod_{l=0}^{3}\left(X-\frac{p^{r}}{64}\sigma^{l}(M)\right)$

where

$\displaystyle L=L(x,w,v,u)=$ $\displaystyle\
x^{3}-50(v^{2}+u^{2})w-125(11v^{2}-4vu-11u^{2}),$ $\displaystyle
M=M(x,w,v,u)=$ $\displaystyle-x^{3}+25x\left(2ux+(7v-u)(v+3u)\right)$
$\displaystyle+125w\left(25w^{2}+10(4v-3u)w+(7v-u)(v+3u)\right)$
$\displaystyle+500(-2v^{3}-3v^{2}u+6vu^{2}+u^{3})$

(see [Hos06, Theorem 1]). Because the exponential quintic Gauss sum
$g_{5r}(5)=\eta_{5r}^{*}(0)$ is one of the roots of $P_{5,5r}^{*}(X)$ which
does not depend on the choice of $\gamma$, we have

$\displaystyle g_{5r}(5)=\frac{p^{r}}{16}L(x,w,v,u)$

(see [Hos06, Corollary 9]). Similarly, for $n=5m$, we have

$\displaystyle g_{5mr}(5)$
$\displaystyle=\frac{p^{mr}}{16}L(x^{(m)},w^{(m)},v^{(m)},u^{(m)}).$

For example, we take $p=11$ and $r=1$. Then we have
$g_{5}(5)=\eta^{*}_{5}(0)=\frac{11}{16}L(1,1,1,0)=-979=-11\cdot 89$. Indeed,
we may check that
$g_{5}(5)=13751\cdot\zeta_{11}^{0}+14730\cdot\sum_{i=1}^{10}\zeta_{11}^{i}=13751-14730=-979$
by the definition using a computer (cf. Section 5 (2) $e=5$).

(3) $e=7$ (cf. Leonard and Williams [LW75], [BEW98, Section 3.9]). Let $p$ be
a prime with $p\equiv 1\ ({\rm mod}\ 7)$. The Jacobi sums $J_{r}(1,1)$ and
$J_{r}(1,2)$ are given by

$\displaystyle J_{r}(1,1)$
$\displaystyle=J(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$
$\displaystyle=\frac{1}{12}\left(Z\zeta_{7}+\sigma^{4}(Z)\zeta_{7}^{2}+\sigma^{5}(Z)\zeta_{7}^{3}+\sigma^{2}(Z)\zeta_{7}^{4}+\sigma(Z)\zeta_{7}^{5}+\sigma^{3}(Z)\zeta_{7}^{6}\right),$
$\displaystyle J_{r}(1,2)$ $\displaystyle=J^{\prime}(t,u)=-t+u\sqrt{-7}$

where $Z=-2x_{1}+6x_{2}+7x_{5}+21x_{6}$,
$\sigma(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})=(x_{1},-x_{3},x_{4},x_{2},(-x_{5}-3x_{6})/2,(x_{5}-x_{6})/2)$
and $x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u\in\mathbb{Z}$ are obtained as the
integer solutions of the system of Diophantine equations

(7)
$\displaystyle\begin{cases}72p^{r}=2x_{1}^{2}+42x_{2}^{2}+42x_{3}^{2}+42x_{4}^{2}+343x_{5}^{2}+1029x_{6}^{2},\\\
12x_{2}^{2}-12x_{4}^{2}+147x_{5}^{2}-441x_{6}^{2}+56x_{1}x_{6}+24x_{2}x_{3}-24x_{2}x_{4}+48x_{3}x_{4}+98x_{5}x_{6}=0,\\\
12x_{3}^{2}-12x_{4}^{2}+49x_{5}^{2}-147x_{6}^{2}+28x_{1}x_{5}+28x_{1}x_{6}+48x_{2}x_{3}+24x_{2}x_{4}+24x_{3}x_{4}\\\
+490x_{5}x_{6}=0,\\\ x_{1}\equiv 1\ ({\rm mod}\ 7),\
(x_{5},x_{6})\neq(0,0),\\\ p^{r}=t^{2}+7u^{2},\ t\equiv 1\ ({\rm mod}\ 7),\
u\equiv 3x_{2}+2x_{3}\ ({\rm mod}\ 7).\end{cases}$

The equations has six solutions
$\sigma^{i}(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})$ $(i=0,1,2,3,4,5)$ which
depend on the choice of $\gamma$. The multiplication matrix $C_{r}$ of the
Gaussian periods $\eta_{r}(0),\ldots,\eta_{r}(6)$ of order $7$ for
$\mathbb{F}_{p^{r}}$ is given by

$\displaystyle C_{r}$
$\displaystyle=\,C_{r}(p,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u)$
$\displaystyle=\left(\begin{array}[]{ccccccc}A-f&B-f&C-f&D-f&E-f&F-f&G-f\\\
B&G&H&I&J&K&H\\\ C&H&F&K&L&L&I\\\ D&I&K&E&J&L&J\\\ E&J&L&J&D&I&K\\\
F&K&L&L&I&C&H\\\ G&H&I&J&K&H&B\\\ \end{array}\right)$

where

$\displaystyle A$ $\displaystyle=\frac{1}{49}(p^{r}+3x_{1}-12t-20),$
$\displaystyle B$
$\displaystyle=\frac{1}{196}(4p^{r}-2x_{1}+28x_{2}-14x_{3}+49x_{5}+49x_{6}+8t+56u-24),$
$\displaystyle C$
$\displaystyle=\frac{1}{98}(2p^{r}-x_{1}+14x_{3}+7x_{4}-49x_{6}+4t+28u-12),$
$\displaystyle D$
$\displaystyle=\frac{1}{196}(4p^{r}-2x_{1}+14x_{2}+28x_{4}-49x_{5}+49x_{6}+8t-56u-24),$
$\displaystyle E$
$\displaystyle=\frac{1}{196}(4p^{r}-2x_{1}-14x_{2}-28x_{4}-49x_{5}+49x_{6}+8t+56u-24),$
$\displaystyle F$
$\displaystyle=\frac{1}{98}(2p^{r}-x_{1}-14x_{3}-7x_{4}-49x_{6}+4t-28u-12),$
$\displaystyle G$
$\displaystyle=\frac{1}{196}(4p^{r}-2x_{1}-28x_{2}+14x_{3}+49x_{5}+49x_{6}+8t-56u-24),$
$\displaystyle H$ $\displaystyle=\frac{1}{147}(3p^{r}+2x_{1}-49x_{5}+6t+3),$
$\displaystyle I$
$\displaystyle=\frac{1}{98}(2p^{r}-x_{1}+7x_{2}+7x_{3}-7x_{4}-10t-14u+2),$
$\displaystyle J$
$\displaystyle=\frac{1}{294}(6p^{r}+4x_{1}+49x_{5}-147x_{6}+12t+6),$
$\displaystyle K$
$\displaystyle=\frac{1}{98}(2p^{r}-x_{1}-7x_{2}-7x_{3}+7x_{4}-10t+14u+2),$
$\displaystyle L$
$\displaystyle=\frac{1}{294}(6p^{r}+4x_{1}+49x_{5}+147x_{6}+12t+6)$

(see [LW75, Theorem] with a typo for $B$ ($147x_{4}$ should be $147x_{5}$)).
Then we have

$\displaystyle P_{7,r}(X)$ $\displaystyle={\rm
Char}_{X}(C_{r}(p,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u)),$ $\displaystyle
P_{7,r}^{*}(X)$
$\displaystyle=P^{*}_{7,r}(p,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u;X)=7^{7}P_{7,r}((X-1)/7)$
$\displaystyle=X^{7}-21p^{r}X^{5}+a_{4}X^{4}+a_{3}X^{3}+a_{2}X^{2}+a_{1}X+a_{0}$

where $a_{i}=a_{i}(p,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u)$ $(0\leq i\leq
4)$ can be obtained explicitly (we omit the display here).

By Theorem 1.4, we get

$\displaystyle
C_{nr}=(-1)^{n-1}C_{r}(p,x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},t,u)^{(n)}=C_{r}(p^{n},x_{1}^{(n)},x_{2}^{(n)},x_{3}^{(n)},x_{4}^{(n)},x_{5}^{(n)},x_{6}^{(n)},t^{(n)},u^{(n)})$

where $x_{1}^{(n)}$, $x_{2}^{(n)}$, $x_{3}^{(n)}$, $x_{4}^{(n)}$,
$x_{5}^{(n)}$, $x_{6}^{(n)}$, $t^{(n)}$, $u^{(n)}$ satisfy the equation (7)
with respect to $p^{nr}$ instead of $p^{r}$. In particular, we have
$J_{nr}(1,1)=J(x_{1}^{(n)},x_{2}^{(n)},x_{3}^{(n)},x_{4}^{(n)},x_{5}^{(n)},x_{6}^{(n)})$,
$J_{nr}(1,2)=J^{\prime}(t^{(n)},u^{(n)})$ and

$\displaystyle p^{nr}$
$\displaystyle=\left(\frac{2x_{1}^{2}+42x_{2}^{2}+42x_{3}^{2}+42x_{4}^{2}+343x_{5}^{2}+1029x_{6}^{2}}{72}\right)^{n}$
$\displaystyle=\frac{2(x_{1}^{(n)})^{2}+42(x_{2}^{(n)})^{2}+42(x_{3}^{(n)})^{2}+42(x_{4}^{(n)})^{2}+343(x_{5}^{(n)})^{2}+1029(x_{6}^{(n)})^{2}}{72},$
$\displaystyle p^{nr}$
$\displaystyle=(t^{2}+7u^{2})^{n}=(t^{(n)})^{2}+7(u^{(n)})^{2}.$

For $n=2$, we get

$\displaystyle x_{1}^{(2)}$
$\displaystyle=\frac{1}{12}(-2x_{1}^{2}+42x_{2}^{2}+42x_{3}^{2}+42x_{4}^{2}-343x_{5}^{2}-1029x_{6}^{2}),$
$\displaystyle x_{2}^{(2)}$
$\displaystyle=\frac{1}{12}(-4x_{1}x_{2}+14x_{2}x_{5}-42x_{3}x_{5}-42x_{4}x_{5}-42x_{2}x_{6}-42x_{3}x_{6}+42x_{4}x_{6}),$
$\displaystyle x_{3}^{(2)}$
$\displaystyle=\frac{1}{12}(-4x_{1}x_{3}-42x_{2}x_{5}-28x_{3}x_{5}-42x_{2}x_{6}-84x_{4}x_{6}),$
$\displaystyle x_{4}^{(2)}$
$\displaystyle=\frac{1}{12}(-4x_{1}x_{4}-42x_{2}x_{5}+14x_{4}x_{5}+42x_{2}x_{6}-84x_{3}x_{6}+42x_{4}x_{6}),$
$\displaystyle x_{5}^{(2)}$
$\displaystyle=\frac{1}{168}(-12x_{2}^{2}+72x_{2}x_{3}+24x_{3}^{2}+72x_{2}x_{4}-12x_{4}^{2}-56x_{1}x_{5}+49x_{5}^{2}-882x_{5}x_{6}-147x_{6}^{2}),$
$\displaystyle x_{6}^{(2)}$
$\displaystyle=\frac{1}{168}(12x_{2}^{2}+24x_{2}x_{3}-24x_{2}x_{4}+48x_{3}x_{4}-12x_{4}^{2}-147x_{5}^{2}-56x_{1}x_{6}-98x_{5}x_{6}+441x_{6}^{2}),$
$\displaystyle t^{(2)}$ $\displaystyle=t^{2}-7u^{2},$ $\displaystyle u^{(2)}$
$\displaystyle=2tu.$

For example, we take $p=29=ef+1$ with $f=4$. Then we get

$\displaystyle P_{7,7}^{*}(X)=$ $\displaystyle\
(X-442569)(X-408465)(X-233682)(X+182671)$
$\displaystyle\cdot(X+259405)(X+317869)(X+324771).$

Indeed, we may check that
$g_{7}(7)=594516413\cdot\zeta_{29}^{0}+594834282\cdot\sum_{i=1}^{28}\zeta_{29}^{i}=594516413-594834282=-317869=-29\cdot
97\cdot 113$ by the definition using a computer (cf. Section 5 (3) $e=7$).

(4) $e=11$, $e=13$ and $e=17$. By using Thaine’s formula [Tha04, page 259], we
can obtain the multiplication matrix $C_{1}$ of the Gaussian periods
$\eta_{1}(0),\ldots,\eta_{1}(e-1)$ of degree $e$ for $\mathbb{F}_{p^{1}}$. By
using Theorem 1.4 as in the case of $e=7$, we get $C_{e}=C_{1}^{(e)}$,
$P_{e,e}(X)={\rm Char}_{X}(C_{1}^{(e)})$ and the explicit factorization of
$P_{e,e}^{*}(X)$ into $e$ linear factors. For example, we take $p=23=ef+1$
with $e=11$ and $f=2$. Then, we get $C_{11}=C_{1}^{(11)}$, $P_{11,11}(X)={\rm
Char}_{X}(C_{1}^{(11)})$ and

$\displaystyle P_{11,11}^{*}(X)=$ $\displaystyle\
(X-166665038)(X-142959444)(X-52918009)(X-47121273)$
$\displaystyle\cdot(X-3199967)(X+19592803)(X+23093817)(X+44439427)$
$\displaystyle\cdot(X+58652208)(X+64390754)(X+202694722)$

with a root $g_{11}(11)=\eta^{*}_{11}(0)=52918009=23\cdot 53\cdot 43441$ (cf.
Section 5 (4) $e=11$). Similarly, by using Theorem 1.4, we can obtain for
$p=53=ef+1$ with $e=13$ and $f=4$, $P_{13,13}^{*}(X)$ with a root
$g_{13}(13)=\eta^{*}_{13}(0)=782475795674=2\cdot 53\cdot 7381847129$ and for
$p=103=ef+1$ with $e=17$ and $f=6$, $P_{17,17}^{*}(X)$ with a root
$g_{17}(17)=\eta^{*}_{17}(0)=-651513206543247755=-5\cdot 7\cdot 103\cdot
172709\cdot 1046412659$ (cf. Section 5 (5) $e=13$, (6) $e=17$).

(5) $e=19$ and $e=23$. We take $p=191=ef+1$ with $e=19$ and $f=10$. As in the
case (4), we can get $C_{19}=C_{1}^{(19)}$, $P_{19,19}(X)={\rm
Char}_{X}(C_{1}^{(19)})$ and the explicit factorization of $P_{19,19}^{*}(X)$
into $19$ linear factors. We see that $\eta^{*}_{19}(i)=p\,\xi_{i}$ and the
$\xi_{i}$’s $(i\neq 0)$ are permuted under the action
$\zeta_{p}\mapsto\zeta_{p}^{\gamma}$ with
$\mathbb{F}_{p}^{\times}=\langle\gamma\rangle$ (which depends on the choice of
$\gamma$). Because we see that $\xi_{0}\in\mathbb{F}_{191}^{\times}$ is of
order $10$ and $\xi_{i}\in\mathbb{F}_{191}^{\times}$ $(1\leq i\leq 18)$ is of
order $190$, we can find
$p\,\xi_{0}=g_{19}(19)=\eta^{*}_{19}(0)=2801935824159299141695=5\cdot 191\cdot
509\cdot 26374987\cdot 218546963$. Similarly, for $p=47=ef+1$ with $e=23$ and
$f=2$, we get $P^{*}_{23,23}(X)$ with a root
$g_{23}(23)=\eta^{*}_{23}(0)=-492643134044787602=-2\cdot 17\cdot 43\cdot
47\cdot 7169472509893$.

We give GAP ([GAP]) computations for examples above. The function ${\tt
MultMat}(e,p,\gamma)$ returns the multiplication matrix $C_{1}$ of the
Gaussian periods $\eta_{1}(0),\ldots,\eta_{1}(e-1)$ of degree $e$ for
$\mathbb{F}_{p^{1}}$ with respect to the generator $\gamma$ of
$\mathbb{F}_{p}^{\times}$ using Thaine’s formula [Tha04, page 259]. The
function ${\tt dComp}(A,B,d)$ returns the $d$-composition
$A\overset{d}{\ast}B$ for two matrices $A$ and $B$.

    MultMat:=function(e,p,g)
      local f,mat,j;
      f:=(p-1)/e;
      mat:=List([0..e-1],i->List([0..e-1],j->(-1/e^2)*Sum([0..e-1],
        l->Sum([0..e],k->Binomial(f*k,f*l)*g^(f*(l*i-k*j)))) mod p));
      if IsEvenInt(f) then for j in [1..e] do mat[1,j]:=mat[1,j]-f;od;
        else for j in [1..e] do mat[e/2+1,j]:=mat[e/2+1,j]-f;od;
      fi;
      return mat;
    end;

    Mode:=function(a,e)
      if a mod e = 0 then return e; else return a mod e;fi;
    end;

    dComp:=function(A,B,d)
      local s,t,i,j,e,mat;
      if Size(A)=Size(B) then e:=Size(A); else return "Input error";
      fi;
      mat:=List([0..e-1],i->List([0..e-1],j->Sum([0..e-1],
        s->Sum([0..e-1],t->A[s+1,t+1]*B[Mode(d*s+i+1,e),Mode(d*t+j+1,e)]))));
      return mat;
    end;

    gap> PrimitiveRootMod(7); # g=3
    3
    gap> C:=MultMat(3,7,3); # Multiplication matrix C1 for e=3, p=7, g=3
    [ [ -2, -2, -1 ],
      [ 0, 1, 1 ],
      [ 1, 1, 0 ] ]
    gap> C2:=dComp(C,C,-1); # C2=C^(2)
    [ [ 10, 11, 12 ],
      [ -5, -4, -7 ],
      [ -4, -7, -5 ] ]
    gap> C3:=dComp(C2,C,-1); # C3=C^(3)
    [ [ -79, -72, -78 ],
      [ 42, 36, 36 ],
      [ 36, 36, 42 ] ]
    gap> P3:=CharacteristicPolynomial(C3); # P3 is the priod polynomial for r=3
    x_1^3+x_1^2-114*x_1+216
    gap> R3:=RootsOfPolynomial(P3); # roots of P3
    [ 9, 2, -12 ]
    gap> L3:=List(R3,x->3*x+1); # roots of the reduced period polynomial P3^*
    [ 28, 7, -35 ]
    gap> X3:=L3/7;
    [ 4, 1, -5 ]
    gap> List(X3,x->x mod 7); # X3[2]=1 mod 7
    [ 4, 1, 2 ]
    gap> List(X3,x->x^2 mod 7);
    [ 2, 1, 4 ]
    gap> List(X3,x->x^3 mod 7); # X3[i] (i<>2) is of order 3 in F7^x
    [ 1, 1, 1 ]
    gap> L3[2]; # L3[2] is the exponential Gauss sum g_3(3)
    7

    gap> PrimitiveRootMod(11); # g=2
    2
    gap> C:=MultMat(5,11,2); # Multiplication matrix C1 for e=5, p=11, g=2
    [ [ -2, -1, -2, -2, -2 ],
      [ 1, 0, 0, 1, 0 ],
      [ 0, 0, 0, 1, 1 ],
      [ 0, 1, 1, 0, 0 ],
      [ 0, 0, 1, 0, 1 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C5:=dComp(C4,C,-1); # C5=C^(5)
    [ [ -25721, -25790, -25680, -25830, -25820 ],
      [ 6420, 6390, 6500, 6400, 6500 ],
      [ 6530, 6500, 6380, 6400, 6400 ],
      [ 6380, 6400, 6400, 6530, 6500 ],
      [ 6390, 6500, 6400, 6500, 6420 ] ]
    gap> P5:=CharacteristicPolynomial(C5); # P5 is the priod polynomial for r=5
    x_1^5+x_1^4-64420*x_1^3-2589700*x_1^2+558588000*x_1+11695320000
    gap> R5:=RootsOfPolynomial(P5); # roots of P5
    [ 255, 90, -20, -130, -196 ]
    gap> L5:=List(R5,x->5*x+1); # roots of the reduced period polynomial P5^*
    [ 1276, 451, -99, -649, -979 ]
    gap> X5:=L5/11;
    [ 116, 41, -9, -59, -89 ]
    gap> List(X5,x->x mod 11); # X5[5]=-1 mod 11
    [ 6, 8, 2, 7, 10 ]
    gap> List(X5,x->x^2 mod 11);
    [ 3, 9, 4, 5, 1 ]
    gap> List(X5,x->x^5 mod 11); # X5[i] (i<>5) is of order 10 in F11^x
    [ 10, 10, 10, 10, 10 ]
    gap> L5[5]; # L5[5] is the exponential Gauss sum g_5(5)
    -979
    gap> Factors(L5[5]);
    [ -11, 89 ]

    gap> PrimitiveRootMod(29); # g=2
    2
    gap> C:=MultMat(7,29,2); # Multiplication matrix C1 for e=7, p=29, g=2
    [ [ -4, -3, -4, -4, -2, -4, -4 ],
      [ 1, 0, 1, 0, 0, 1, 1 ],
      [ 0, 1, 0, 1, 1, 1, 0 ],
      [ 0, 0, 1, 2, 0, 1, 0 ],
      [ 2, 0, 1, 0, 0, 0, 1 ],
      [ 0, 1, 1, 1, 0, 0, 1 ],
      [ 0, 1, 0, 0, 1, 1, 1 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C3:=dComp(C2,C,-1);; # C3=C^(3)
    gap> C7:=dComp(C4,C3,-1);; # C7=C^(7)
    gap> P7:=CharacteristicPolynomial(C7);; # P7 is the priod polynomial for r=7
    gap> R7:=RootsOfPolynomial(P7);; # roots of P7
    gap> L7:=List(R7,x->7*x+1); # roots of the reduced period polynomial P7^*
    [ 442569, 408465, 233682, -182671, -259405, -317869, -324771 ]
    gap> X7:=L7/29;
    [ 15261, 14085, 8058, -6299, -8945, -10961, -11199 ]
    gap> List(X7,x->x mod 29); # X7[6]=1 mod 29
    [ 7, 20, 25, 23, 16, 1, 24 ]
    gap> List(X7,x->x^2 mod 29);
    [ 20, 23, 16, 7, 24, 1, 25 ]
    gap> List(X7,x->x^4 mod 29);
    [ 23, 7, 24, 20, 25, 1, 16 ]
    gap> List(X7,x->x^7 mod 29); # X7[i] (i<>6) is of order 7 in F29^x
    [ 1, 1, 1, 1, 1, 1, 1 ]
    gap> L7[6]; # L7[6] is the exponential Gauss sum g_7(7)
    -317869
    gap> Factors(L7[6]);
    [ -29, 97, 113 ]

    gap> PrimitiveRootMod(23); # g=5
    5
    gap> C:=MultMat(11,23,5); # Multiplication matrix C1 for e=11, p=23, g=5
    [ [ -2, -2, -1, -2, -2, -2, -2, -2, -2, -2, -2 ],
      [ 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0 ],
      [ 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ],
      [ 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ],
      [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0 ],
      [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0 ],
      [ 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C8:=dComp(C4,C4,-1);; # C8=C^(8)
    gap> C3:=dComp(C2,C,-1);; # C3=C^(3)
    gap> C11:=dComp(C8,C3,-1);; # C11=C^(11)
    gap> P11:=CharacteristicPolynomial(C11);; # P11 is the priod polynomial for r=11
    gap> R11:=RootsOfPolynomial(P11);; # roots of P11
    gap> L11:=List(R11,x->11*x+1); # roots of the reduced period polynomial P11^*
    [ 166665038, 142959444, 52918009, 47121273, 3199967, -19592803,
      -23093817, -44439427, -58652208, -64390754, -202694722 ]
    gap> X11:=L11/23;;
    gap> List(X11,x->x mod 23); # X11[3]=1 mod 23
    [ 18, 16, 1, 3, 2, 13, 9, 12, 6, 8, 4 ]
    gap> List(X11,x->x^2 mod 23);
    [ 2, 3, 1, 9, 4, 8, 12, 6, 13, 18, 16 ]
    gap> List(X11,x->x^11 mod 23); # X13[i] (i<>3) is of order 11 in F23^x
    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
    gap> L11[3]; # L11[3] is the exponential Gauss sum g_{11}(11)
    52918009
    gap> Factors(L11[3]);
    [ 23, 53, 43411 ]

    gap> PrimitiveRootMod(53); # g=2
    2
    gap> C:=MultMat(13,53,2); # Multiplication matrix C1 for e=13, p=53, g=2
    [ [ -4, -3, -4, -4, -4, -4, -4, -2, -4, -4, -4, -4, -4 ],
      [ 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1 ],
      [ 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1 ],
      [ 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1 ],
      [ 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 1, 0 ],
      [ 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0 ],
      [ 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0 ],
      [ 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0 ],
      [ 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C8:=dComp(C4,C4,-1);; # C8=C^(8)
    gap> C5:=dComp(C4,C,-1);; # C5=C^(5)
    gap> C13:=dComp(C8,C5,-1);; # C13=C^(13)
    gap> P13:=CharacteristicPolynomial(C13);; # P13 is the priod polynomial for r=13
    gap> R13:=RootsOfPolynomial(P13) # roots of P13
    gap> L13:=List(R13,x->13*x+1); # roots of the reduced period polynomial P13^*
    [ 1040615291340, 782475795674, 664438112586, 338244988654, 117899008800,
      83828569254, -186980700750, -238169301889, -245670171356, -277653262665,
      -427932303889, -740552966334, -910543059425 ]
    gap> X13:=L13/53;;
    gap> List(X13,x->x mod 53); # X13[2]=1 mod 53
    [ 47, 1, 49, 16, 13, 15, 42, 36, 44, 46, 10, 28, 24 ]
    gap> List(X13,x->x^2 mod 53);
    [ 36, 1, 16, 44, 10, 13, 15, 24, 28, 49, 47, 42, 46 ]
    gap> List(X13,x->x^4 mod 53);
    [ 24, 1, 44, 28, 47, 10, 13, 46, 42, 16, 36, 15, 49 ]
    gap> List(X13,x->x^13 mod 53); # X13[i] (i<>2) is of order 13 in F53^x
    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
    gap> L13[2]; # L13[2] is the exponential Gauss sum g_{13}(13)
    782475795674
    gap> Factors(L13[2]);
    [ 2, 53, 7381847129 ]

    gap> PrimitiveRootMod(103); # g=5
    5
    gap> C:=MultMat(17,103,5); # Multiplication matrix C1 for e=17, p=103, g=5
    [ [ -4, -6, -6, -6, -6, -6, -6, -6, -6, -6, -5, -4, -6, -6, -6, -6, -6 ],
      [ 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1 ],
      [ 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1 ],
      [ 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0 ],
      [ 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0 ],
      [ 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2 ],
      [ 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0 ],
      [ 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0 ],
      [ 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ],
      [ 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 0, 0, 0 ],
      [ 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0 ],
      [ 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0 ],
      [ 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ],
      [ 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1 ],
      [ 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C8:=dComp(C4,C4,-1);; # C8=C^(8)
    gap> C16:=dComp(C8,C8,-1);; # C16=C^(16)
    gap> C17:=dComp(C16,C,-1);; # C17=C^(17)
    gap> P17:=CharacteristicPolynomial(C17);; # P17 is the priod polynomial for r=17
    gap> R17:=RootsOfPolynomial(P17);; # roots of P17
    gap> L17:=List(R17,x->17*x+1); # roots of the reduced period polynomial P17^*
    [ 670357206530506901, 670088231006862759, 587253242462231659,
      464742031061114921, 461908111585063663, 356621718684896633,
      282238003107978403, 163898849457734107, 35922811461007315,
      -197783211402587952, -205052440856501077, -243310155546790559,
      -289516205265127375, -373934090375919493, -478731856802195967,
      -651513206543247755, -1253189038565026183 ]
    gap> X17:=L17/103;;
    gap> List(X17,x->x mod 103); # X17[16]=-1 mod 103
    [ 73, 31, 24, 94, 90, 22, 3, 95, 89, 42, 10, 80, 27, 69, 39, 102, 37 ]
    gap> List(X17,x->x^2 mod 103);
    [ 76, 34, 61, 81, 66, 72, 9, 64, 93, 13, 100, 14, 8, 23, 79, 1, 30 ]
    gap> List(X17,x->x^3 mod 103);
    [ 89, 24, 22, 95, 69, 39, 27, 3, 37, 31, 73, 90, 10, 42, 94, 102, 80 ]
    gap> List(X17,x->x^17 mod 103); # X17[i] (i<>16) is of order 34 in F103^x
    [ 102, 102, 102, 102, 102, 102, 102, 102, 102, 102,
      102, 102, 102, 102, 102, 102, 102 ]
    gap> L17[16]; # L17[16] is the exponential Gauss sum g_{17}(17)
    -651513206543247755
    gap> Factors(L17[16]);
    [ -5, 7, 103, 172709, 1046412659 ]

    gap> PrimitiveRootMod(191); # g=19
    19
    gap> C:=MultMat(19,191,19); # Multiplication matrix C1 for e=19, p=191, g=19
    [ [ -10, -10, -10, -10, -10, -10, -9, -8, -8, -10,
        -10, -8, -10, -10, -10, -10, -10, -10, -8 ],
      [ 0, 2, 0, 1, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 1, 0, 1 ],
      [ 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 2 ],
      [ 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2 ],
      [ 0, 2, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0 ],
      [ 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0 ],
      [ 2, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0 ],
      [ 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0 ],
      [ 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 2, 1, 1, 0, 0, 0, 1 ],
      [ 0, 1, 1, 2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0 ],
      [ 2, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 2, 0, 0, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 1, 0, 0 ],
      [ 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1 ],
      [ 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 1, 1 ],
      [ 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0 ],
      [ 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0 ],
      [ 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 0 ],
      [ 2, 0, 1, 2, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C8:=dComp(C4,C4,-1);; # C8=C^(8)
    gap> C16:=dComp(C8,C8,-1);; # C16=C^(16)
    gap> C3:=dComp(C2,C,-1);; # C3=C^(3)
    gap> C19:=dComp(C16,C3,-1);; # C19=C^(19)
    gap> P19:=CharacteristicPolynomial(C19);; # P19 is the priod polynomial for r=19
    gap> R19:=RootsOfPolynomial(P19);; # roots of P19
    gap> L19:=List(R19,x->19*x+1); # roots of the reduced period polynomial P19^*
    [ 55891098112086637001228, 21343147495425176673226, 16127550524178031129657,
      14355859672843887131634, 10195021892556248415182, 7777342710886644977131,
      5776338119599847350627, 5080513863740739683465, 2801935824159299141695,
      859413598509266105572, -1967831693815607448660, -2042500136091280335075,
      -5599389538599795630810, -11060282774339943468556, -14117536712596171711328,
      -19950229182831388897609, -27250892079645375357179, -28187266231514473770821,
      -30032293464551740989379 ]
    gap> X19:=List(L19,x->x/191);;
    gap> List(X19,x->x mod 191);
    [ 58, 182, 157, 178, 94, 21, 126, 151, 142, 57,
      99, 113, 146, 88, 112, 143, 105, 137, 183 ]
    gap> List(X19,x->x^10 mod 191); # X19[9]^10=1 mod 191
    [ 52, 136, 107, 121, 125, 30, 25, 154, 1, 32,
      36, 69, 6, 160, 5, 150, 153, 177, 180 ]
    gap> List(X19,x->x^2 mod 191);
    [ 117, 81, 10, 169, 50, 59, 23, 72, 109, 2,
      60, 163, 115, 104, 129, 12, 138, 51, 64 ]
    gap> List(X19,x->x^5 mod 191); # X19[9] is of order 10 in F191^x
    [ 166, 161, 38, 11, 70, 139, 186, 66, 190, 37,
      185, 159, 31, 55, 14, 155, 41, 122, 84 ]
    gap> List(X19,x->x^19 mod 191); # X19[i] (i<>9) is of order 190 in F191^x
    [ 152, 152, 152, 152, 152, 152, 152, 152, 152, 152,
      152, 152, 152, 152, 152, 152, 152, 152, 152 ]
    gap> L19[9]; # L19[9] is the exponential Gauss sum g_{19}(19)
    2801935824159299141695
    gap> Factors(L19[9]);
    [ 5, 191, 509, 26374987, 218546963 ]

    gap> PrimitiveRootMod(47); # g=5
    5
    gap> C:=MultMat(23,47,5);  # Multiplication matrix C1 for e=23, p=47, g=5
    [ [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2,
        -2, -2, -2, -2, -2, -2, -1, -2, -2, -2, -2 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
      [ 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ],
      [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
      [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
      [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],
      [ 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
      [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ]
    gap> C2:=dComp(C,C,-1);; # C2=C^(2)
    gap> C4:=dComp(C2,C2,-1);; # C4=C^(4)
    gap> C8:=dComp(C4,C4,-1);; # C8=C^(8)
    gap> C16:=dComp(C8,C8,-1);; # C16=C^(16)
    gap> C3:=dComp(C2,C,-1);; # C3=C^(3)
    gap> C7:=dComp(C4,C3,-1);; # C7=C^(7)
    gap> C23:=dComp(C16,C7,-1);; # C23=C^(23)
    gap> P23:=CharacteristicPolynomial(C23);; # P23 is the priod polynomial for r=23
    gap> R23:=RootsOfPolynomial(P23);; # roots of P23
    gap> L23:=List(R23,x->23*x+1); # roots of the reduced period polynomial P23^*
    [ 142339874433137221525, 118065170266710759348, 90401156916499269233,
      55373954393947818396, 55099193646218848063, 42654144441633168738,
      42378310496086559486, 36268843595424974262, 35660322726333362220,
      34760976326466677323, 28446187386897694871, 17050560055492972666,
      -492643134044787602, -9055501540645768832, -16107397702852877550,
      -31331987537967805455, -36858108220907188977, -38922282154313258582,
      -39922556198217904917, -67269172064831016965, -90222434992270940059,
      -151942428479066503710, -216374182659731273482 ]
    gap> X23:=L23/47;;
    gap> List(X23,x->x mod 47); # X23[13]=1 mod 47
    [ 9, 32, 24, 4, 17, 12, 8, 2, 3, 37, 14, 18,
      1, 21, 36, 16, 25, 28, 42, 27, 34, 6, 7 ]
    gap> List(X23,x->x^2 mod 47);
    [ 34, 37, 12, 16, 7, 3, 17, 4, 9, 6, 8, 42,
      1, 18, 27, 21, 14, 32, 25, 24, 28, 36, 2 ]
    gap> List(X23,x->x^23 mod 47); # X23[i] (i<>13) is of order 23 in F47^x
    [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
    gap> L23[13]; # L23[13] is the exponential Gauss sum g_{23}(23)
    -492643134044787602
    gap> Factors(L23[13]);
    [ -2, 17, 43, 47, 7169472509893 ]

######  Acknowledgment.

The authors would like to thank one of the referees who read the manuscript
very carefully and gave helpful suggestions.

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