# Quadratic number fields with unramified $SL_{2}(5)$-extensions

Joachim König  Korea National University of Education, Department of
Mathematics Education, 28173 Cheongju, South Korea

###### Abstract.

Continuing the line of thought of an earlier work [8], we provide the first
infinite family of quadratic number fields with everywhere unramified Galois
extensions of Galois group $SL_{2}(5)$, the (unique) smallest nonsolvable
group for which this problem was previously open. Our approach also improves
upon [8] by yielding the first infinite family of real-quadratic fields
possessing an unramified Galois extension whose Galois group is perfect and
not generated by involutions. Our result also amounts to a new existence
result on quintic number fields with squarefree discriminant and additional
local conditions.

##  1\. Introduction and main result

This work continues the investigations of [8], motivated by the following
folklore conjecture in inverse Galois theory.

######  Conjecture 1.

For any finite group $G$, there exist infinitely many quadratic number fields
$K$ such that $K$ possesses a Galois extension with Galois group $G$
unramified at all (finite and infinite) primes of $K$.

Stronger versions of Conjecture 1 also take into account the asymptotic
distribution of the number of unramified extensions (when the quadratic fields
are counted by discriminant), generalizing the Cohen-Lenstra heuristics; see
in particular [17]. Conjecture 1 is known to be true only in some special
cases, including the case of cyclic groups (where Conjecture 1 relates
directly to the “class number divisibility problem” of class field theory, cf.
[1]), symmetric and alternating groups (e.g., [18] (which does not consider
ramification at infinity) or [6] (which does)), and a few other almost simple
finite groups (e.g., [10]). Groups such as $G=A_{n}$, and more generally,
perfect groups $G$ (that is, groups with trivial abelianization), are in some
way on the opposite side of abelian groups, and investigating Conjecture 1 for
them may be seen as complementary to the class number divisibility problem.
While the above nonsolvable examples share the property that they are
generated by involutions, [8] obtained a solution for the first cases of
perfect groups not generated by involutions (a property making certain direct
approaches via, e.g., Abhyankar’s lemma impossible), including the group
$SL_{2}(7)$.

In this paper, we solve the smallest open non-solvable case of the conjecture,
namely:

######  Theorem 1.1.

There exist infinitely many imaginary-quadratic and infinitely many real-
quadratic number fields possessing an everywhere (i.e., including the
archimedean primes) unramified Galois extension with Galois group $SL_{2}(5)$.

We will see that the proof of this result about $G=SL_{2}(5)$ is significantly
more intricate than the one of the (known) analogous result for $A_{5}(\cong
PSL_{2}(5))$; progressing from one to the other corresponds to a central
embedding problem with kernel $C_{2}$ and certain extra conditions; those
conditions however make the solution rather nontrivial. On a similar note,
while $A_{5}$ has been known to be a Galois group over $\mathbb{Q}$ since the
19th century, $SL_{2}(5)$ was first shown to be realizable by Sonn [16] in
1980.

Note also that while the proof of Theorem 1.1 shares some common theoretical
ideas with the previously obtained results in [8] for the two perfect groups
$G=SL_{2}(7)$ and $G=2.A_{7}$, it requires a significant amount of extra
computational effort compared to these. On the other hand, the assertion on
real-quadratic number fields in Theorem 1.1 goes beyond what was obtained for
the groups considered in [8]; indeed, the construction there automatically
lead only to imaginary-quadratic fields. We will elaborate in the beginning of
Section 3 on the additional difficulty of the case $G=SL_{2}(5)$ compared to
the previously solved cases. In Corollary 3.3, we give a fully explicit family
of quadratic number fields fulfilling the assertion of Theorem 1.1. This also
yields a lower bound on the asymptotic distribution of such quadratic fields,
although obviously one far away from the actual expectation of the generalized
Cohen-Lenstra heuristics. Magma code for the verification of several claims
about computations is contained in the ancillary file sl25_unram_anc.txt.

Finally, the reader may compare works such as [13] and [4], with which our
result shares the common idea of “building on top” of an initial nonsolvable
unramified extension (although with the notable difference that those papers
are not focussed on realizing an extension with a particular Galois group, and
instead aim at the construction of infinite towers).

##  2\. Prerequisites

We recall some key results on specializations of function field extensions and
on central embedding problems. This section is largely identical with the
analogous section in [8], and may be skipped by a reader familiar with the
techniques.

###  2.1. Local behavior of specializations of function field extensions

For a finite Galois extension $N/K(t)$ of function fields and any value
$t_{0}\in\mathbb{P}^{1}(K)$, we denote by $N_{t_{0}}/K$ the specialization of
$N/K(t)$ at $t\mapsto t_{0}$, i.e., the residue extension at any place
extending $t\mapsto t_{0}$ in $N$. Recall that $N/K(t)$ is called $K$-regular
if $K$ is algebraically closed in $N$. The following well-known theorem, cf.
[2, Prop. 4.2], ties the ramification in specializations of a function field
extension to the inertia groups at branch points of that function field
extension.

######  Theorem 2.1.

Let $K$ be a number field and $N/K(t)$ a finite $K$-regular Galois extension
with Galois group $G$. Then there exists a finite set $\mathcal{S}_{0}$ of
primes, depending only on $N/K(t)$, such that the following holds for every
prime $\mathfrak{p}$ of $K$ not in $\mathcal{S}_{0}$:
If $t_{0}\in K$ is not a branch point of $N/K(t)$ then the following condition
is necessary for $\mathfrak{p}$ to be ramified in the specialization
$N_{t_{0}}/K$:

$e_{i}:=I_{\mathfrak{p}}(t_{0},t_{i})>0\text{ for some (automatically unique,
up to algebraic conjugates) branch point $t_{i}$.}$

Here $I_{\mathfrak{p}}(t_{0},t_{i})$ is the intersection multiplicity of
$t_{0}$ and $t_{i}$ at the prime $\mathfrak{p}$. Furthermore, the inertia
group of a prime extending $\mathfrak{p}$ in the specialization $N_{t_{0}}/K$
is then conjugate in $G$ to $\langle\tau^{e_{i}}\rangle$, where $\tau$ is a
generator of an inertia subgroup over the branch point $t\mapsto t_{i}$ of
$K(t)$.

Regarding the definition of intersection multiplicity $I_{\mathfrak{p}}$
occurring in Theorem 2.1, note that in the special case $K=\mathbb{Q}$ this
may be defined conveniently in the following way: Let $f(X)\in{\mathbb{Z}}[X]$
be the irreducible polynomial of $t_{i}$ over ${\mathbb{Z}}$ and
$\tilde{f}(X,Y)$ its homogenization (with $\tilde{f}:=Y$ in the special case
$t_{i}=\infty)$. Let $t_{0}=\frac{a}{b}$ with $a,b\in{\mathbb{Z}}$ coprime,
and let $p$ be a prime number. Then $I_{p}(t_{0},t_{i})$ is the multiplicity
of $p$ in $\tilde{f}(a,b)$.

The following result (cf. [9, Thm. 4.1]) clarifies the structure of
decomposition groups in specializations. To state it, we need to introduce a
bit more notation. Given a finite Galois extension $N/K(t)$ with group $G$ and
a $K$-rational place $t\mapsto a\in\mathbb{P}^{1}(K)$, denote by $D_{a}$ and
$I_{a}$ the decomposition and inertia group at the place $t\mapsto a$
respectively. Furthermore, given a prime $p$ of $K$, denote by $D_{a,p}$ the
decomposition group at $p$ in the residue extension $N_{a}/K$. Note that the
Galois group of $N_{a}/K$ is identified with $D_{a}/I_{a}$, and hence
$D_{a,p}$ is identified with a subgroup of this group.

######  Theorem 2.2.

Let $K$ be a number field and $N/K(t)$ a finite $K$-regular Galois extension
with Galois group $G$. Let $t\mapsto t_{i}\in\mathbb{P}^{1}(\overline{K})$ be
a branch point of $N/K(t)$, and let $I_{t_{i}}$ and $D_{t_{i}}$ denote the
inertia and decomposition group at $t\mapsto t_{i}$ in $N(t_{i})/K(t_{i})(t)$.
Then there exists a finite set $\mathcal{S}_{1}$ of primes of $K$, depending
only on $N/K(t)$ and containing the set $\mathcal{S}_{0}$ from Theorem 2.1,
such that for all primes $\mathfrak{p}$ of $K$ not in $\mathcal{S}_{1}$ and
all non-branch points $a\in\mathbb{P}^{1}(K)$ of $N/K(t)$, the following hold:

  * a)

Assume that $\mathfrak{p}$ ramifies in $N_{a}/K$ and
$I_{\mathfrak{p}}(a,t_{i})>0$.111The latter is guaranteed to happen for some
branch point $t_{i}$, unique up to algebraic conjugates, by Theorem 2.1. Let
$D_{t_{i},\mathfrak{p}^{\prime}}$ denote the decomposition group of
$N(t_{i})_{t_{i}}/K(t_{i})$ at the (unique) prime $\mathfrak{p}^{\prime}$ of
$K(t_{i})$ extending $\mathfrak{p}$ such that
$I_{\mathfrak{p}^{\prime}}(a,t_{i})>0$. Then the decomposition group
$D_{a,\mathfrak{p}}$ at $\mathfrak{p}$ in $N_{a}/K$ is identified (up to
conjugacy in $G$) with a subgroup of $D_{t_{i}}$, and fulfills
$\varphi(D_{a,\mathfrak{p}})=D_{t_{i},\mathfrak{p}^{\prime}}$, where
$\varphi:D_{t_{i}}\to D_{t_{i}}/I_{t_{i}}$ is the canonical epimorphism.

  * b)

In particular, if additionally $I_{\mathfrak{p}}(a,t_{i})=1$, then
$D_{a,\mathfrak{p}}=\varphi^{-1}(D_{t_{i},\mathfrak{p}^{\prime}})$.

######  Remark 1.

As noted in [11, Theorem 2.2], the exceptional set $\mathcal{S}_{1}$ in
Theorem 2.2 can be effectively bounded from above as follows. If $O_{K}$ is
the ring of integers of $K$ and $f(t,X)\in O_{K}[t,X]$ is a polynomial with
splitting field $N$, then a prime $p$ of $K$ is not in $\mathcal{S}$ as soon
as all of the following are fulfilled: a) $p\not||G|$; furthermore, if
$\Delta\in O_{K}[t]$ denotes the discriminant of $f$ with respect to $x$, then
b) $p$ does not divide the leading coefficient of $\Delta$, and c) the number
of distinct roots of $\Delta$ does not decrease upon reduction modulo $p$.

###  2.2. Embedding problems

We recall some basic terminology around embedding problems. A finite embedding
problem over a field $K$ is a pair $(\varphi:G_{K}\to
G,\varepsilon:\tilde{G}\to G)$, where $\varphi$ is a (continuous) epimorphism
from the absolute Galois group $G_{K}$ of $K$ onto $G$, and $\varepsilon$ is
an epimorphism between finite groups $\tilde{G}$ and $G$ fitting in an exact
sequence $1\to N\to\tilde{G}\to G\to 1$. The kernel $N=\ker(\varepsilon)$ is
called the kernel of the embedding problem. An embedding problem is called
central if $\ker(\varepsilon)\leq Z(\tilde{G})$. A (continuous) homomorphism
$\psi:G_{K}\to\tilde{G}$ is called a solution to $(\varphi,\varepsilon)$ if
the composition $\varepsilon\circ\psi$ equals $\varphi$. In this case, the
fixed field of $\ker(\psi)$ is called a solution field to the embedding
problem. A solution $\psi$ is called a proper solution if it is surjective. In
this case, the field extension of the solution field over $K$ has full Galois
group $\tilde{G}$.

If $K$ is a number field and $\mathfrak{p}$ is a prime of $K$, every embedding
problem $(\varphi,\varepsilon)$ induces an associated local embedding problem
$(\varphi_{\mathfrak{p}},\varepsilon_{\mathfrak{p}})$ defined as follows:
$\varphi_{\mathfrak{p}}$ is the restriction of $\varphi$ to
$G_{K_{\mathfrak{p}}}$ (well defined up to fixing an embedding of
$\overline{K}$ into $\overline{K_{\mathfrak{p}}}$), and
$\varepsilon_{\mathfrak{p}}$ is the restriction of $\varepsilon$ to
$\varepsilon^{-1}(G({\mathfrak{p}}))$, where
$G(\mathfrak{p}):=\varphi_{\mathfrak{p}}(G_{K_{\mathfrak{p}}})$.

We require two well-known results on central embedding problems. The first
(cf. [14, Chapter IV, Cor. 10.2]) is a local global-principle for the
solvability of such embedding problems, which is essentially a consequence of
the local-global principle for Brauer embedding problems over number fields
([14, Chapter IV, Cor. 7.8]).

######  Proposition 2.3.

Let $\Gamma=C.G$ be a central extension of $G$ by a cyclic group $C$ of prime
order and $\varepsilon:\Gamma\to G$ the canonical projection. Let
$\varphi:G_{\mathbb{Q}}\to G$ be a continuous epimorphism. Then the embedding
problem $(\varphi,\varepsilon)$ is solvable if and only if all associated
local embedding problems $(\varphi_{p},\varepsilon_{p})$ are solvable, where
$p$ runs through all primes of $\mathbb{Q}$ (including the infinite one).

######  Remark 2.

If additionally $|C|=2$, the above local-global principle holds already with
the set of all primes replaced by “the set of all primes, with one exception”,
hence, e.g., with the set of all finite primes. This is due to the product
formula for Hasse invariants for Brauer embedding problems.

Since the local embedding problem at an unramified prime is always solvable,
the above lemma gives an efficient method to check for global solvability of a
given embedding problem by investigating only the finitely many ramified
primes.

The second result (see [15, Prop. 2.1.7]) is a tool to control ramification in
solutions of central embedding problems, assuming solvability.

######  Proposition 2.4.

Let $\Gamma=C.G$ be a central extension of $G$ by a finite abelian group $C$,
let $\varepsilon:\Gamma\to G$ be the canonical projection, and let
$\varphi:G_{\mathbb{Q}}\to G$ be a continuous epimorphism such that the
embedding problem $(\varphi,\varepsilon)$ has a solution. For each finite
prime $p$, let $\tilde{\varphi_{p}}:G_{\mathbb{Q}_{p}}\to\Gamma$ be a solution
of the associated local embedding problem $(\varphi_{p},\epsilon_{p})$, chosen
such that all but finitely many $\tilde{\varphi_{p}}$ are unramified. Then
there exists a (not necessarily proper) solution
$\tilde{\varphi}:G_{\mathbb{Q}}\to\Gamma$ of $(\varphi,\epsilon)$ such that
for all finite primes $p$, the restrictions of $\tilde{\varphi}$ and
$\tilde{\varphi_{p}}$ to the inertia group inside $G_{\mathbb{Q}_{p}}$
coincide. In particular, $\tilde{\varphi}$ is ramified exactly at those finite
primes $p$ for which $\tilde{\varphi_{p}}$ is ramified.

##  3\. Proof of Theorem 1.1

###  3.1. Preliminary considerations

The basic approach in the proof of the case $G=SL_{2}(5)$ is the same as for
the realization of $SL_{2}(7)$ in [8]: we try to realize $PGL_{2}(5)\cong
S_{5}$ over $\mathbb{Q}$ in such a way that all nontrivial inertia groups are
generated by involutions outside of $PSL_{2}(5)$ (i.e., in the representation
of $PGL_{2}(5)$ as $S_{5}$, by transpositions), and such that furthermore no
local obstructions to the central embedding problem given by $1\to C_{2}\to
2.S_{5}\to S_{5}\to 1$ arise. Here, $2.S_{5}$ denotes the stem extension of
$S_{5}$ in which the transpositions of $S_{5}$ are split. The centralizer of
such an involution in $S_{5}$ is conjugate to
$\langle(1,2),Sym(\\{3,4,5\\})\rangle\cong C_{2}\times S_{3}$. Since the
subgroup $C_{6}\leq C_{2}\times S_{3}$ of $S_{5}$ is split in the central
extension, the only local obstruction to the embedding problem which can arise
at a tamely ramified prime with inertia group $C_{2}$ (generated by a
transposition) occurs when the decomposition group equals $C_{2}\times C_{2}$.

Our main task is therefore to guarantee the existence of tamely ramified
$S_{5}$-extensions of $\mathbb{Q}$ in which all inertia groups at ramified
primes are generated by transpositions and such that the corresponding
decomposition groups are additionally cyclic (namely, of order $2$ or $6$).
Recalling additionally that tame $S_{n}$-extensions in which all non-trivial
inertia groups at finite primes are generated by transpositions are exactly
those whose degree-$n$ subextension has squarefree discriminant, we thus have
a nice translation of our problem into a combination of two previously-
investigated properties.

######  Lemma 3.1.

Let $F/\mathbb{Q}$ be a degree-$5$ extension with Galois closure
$\Omega/\mathbb{Q}$ fulfilling both of the following:

  * 1)

The discriminant $\Delta(F)$ is squarefree.

  * 2)

$\Omega/\mathbb{Q}$ is “locally cyclic”, i.e., all its decomposition groups
are cyclic.

Then $\Omega/\mathbb{Q}(\sqrt{\Delta(F)})$ embeds into an
$SL_{2}(5)$-extension unramified at all finite primes.

###### Proof.

Condition 1) translates to saying that $F/\mathbb{Q}$ is tamely ramified with
all non-trivial inertia groups generated by transpositions. Since a transitive
subgroup of $S_{n}$ generated by transpositions is necessarily $S_{n}$ itself,
we have $\textrm{Gal}(\Omega/\mathbb{Q})\cong S_{5}$ and
$\textrm{Gal}(\Omega/\mathbb{Q}(\sqrt{\Delta(F)}))\cong A_{5}$, with
$\Omega/\mathbb{Q}(\sqrt{\Delta(F)})$ unramified at all finite primes. As
deduced above, the central embedding problem induced by $1\to C_{2}\to
2.S_{5}\to S_{5}\cong\textrm{Gal}(\Omega/\mathbb{Q})\to 1$ has no local
obstruction at any ramified finite prime, and hence no local obstruction at
any finite prime. Due to Proposition 2.3 and Remark 2, it is therefore
solvable (and automatically properly solvable, since the group extension is
non-split). Using additionally Proposition 2.4 (together with the fact that
all the subgroups generated by transpositions split in the extension
$2.S_{5}$), we see that the solution field $L$ may be chosen such that
$L/\Omega$, and hence even $L/\mathbb{Q}(\sqrt{\Delta(F)})$, is unramified at
all finite primes∎

Number fields $F$ fulfilling part 1) of Lemma 3.1 are indeed well-known for
all degrees, see, e.g., [18] or [6]; for quintics, Bhargava has obtained much
stronger results on the asymptotic distribution of such fields, see in
particular [3, Theorem 1.3]. Also, part 2) was obtained for $S_{5}$ in [7]
(however, explicitly not fulfilling 1) at the same time). In order to get both
properties simultaneously, we search for geometric $S_{5}$-realizations with
suitable local behavior at ramified places. In [8], the analog (with condition
1) suitably adapted) for $G=SL_{2}(7)$ was obtained by using suitable “three
branch point” extensions with group $PGL_{2}(7)$. The main technical tool was
a “pullback” construction which, roughly speaking, allowed to discard two of
the three branch points, while favorably manipulating the residue extension at
the third one.

However, we face one technical problem: while there are of course some very
easy polynomials $f(t,X)$ with Galois group $S_{5}$ over $\mathbb{Q}(t)$ (such
as $f(t,X)=X^{5}-t(X-1)$), the translation process of [8] seems to result in
certain “badly behaving” primes. E.g., for the polynomial $X^{5}-t(X-1)$, the
residue extension at the only branch point $t_{0}$ with inertia group of order
$2$ contains the quadratic field $\mathbb{Q}(\sqrt{-2})$. Since
$\mathbb{Q}(\sqrt{t_{0}})=\mathbb{Q}(\sqrt{5})$, one may “swallow” this
quadratic field via translate $t(s):=-10s^{2}$. However, for the translated
polynomial $f(t(s),X)=X^{5}+10s^{2}(X-1)$, the prime $p=5$ will now ramify of
ramification index $>2$ in all specializations. The same in fact seems to
occur for all three-branch-point extensions with Galois group $S_{5}$. One
therefore needs to dig a bit deeper and search among some 4-branch point
extensions with group $S_{5}$, corresponding to finding suitable members of
families $f(a,t,X)$ of $S_{5}$-polynomials with four branch points (with $a$
to be specialized suitably in the process). Adding extra branch points comes
with the downside of having to control more and more residue extensions at
branch points simultaneously. There is, however, also a notable upside: By a
well-known argument due to Serre (e.g., Example 10.2 in [14, Chapter I.10]), a
three branch point Galois extension of $\mathbb{Q}(t)$ with Galois group $G$
cannot possible specialize to totally real extensions, except for a few small
dihedral groups $G$, meaning that via a specialization approach using such an
extension one cannot hope for real quadratic fields with everywhere (namely,
including the infinite primes) unramified $G$-extensions. Our 4-branch point
approach below will, however, succeed. Theorem 3.2 below identifies quadratic
number fields with an unramified $SL_{2}(5)$-extension inside an explicitly
given infinite family.

###  3.2. A strong version of Theorem 1.1 and its proof

######  Theorem 3.2.

Assume that $w=\frac{w_{1}}{w_{2}}\in\mathbb{Q}$ (with coprime integers
$w_{1},w_{2}$) fulfills all of the following:

  * a)

$(-50w_{1}^{2}+27w_{2}^{2})(10w_{1}^{2}+8w_{1}w_{2}+w_{2}^{2})\in\mathbb{Z}$
is equivalent, up to squares, to a squarefree number all of whose prime
divisors are congruent to $1$ modulo $3$.

  * b)

$w\neq 0$, and $w\in[-0.7348,-0.645]\cup[-0.155,0.7348]$.

Then there exist infinitely many $s\in\mathbb{Q}$ such that the following
holds: If $\Delta_{s}$ denotes the discriminant of the polynomial

$f_{w,s}(X)=X^{2}(X-1)^{3}+2s^{2}(50w^{2}-27)(10w^{2}+8w+1)(X-2w^{2}),$

then the quadratic number field $\mathbb{Q}(\sqrt{\Delta_{s}})$ possesses an
everywhere (i.e., including the infinite primes) unramified
$SL_{2}(5)$-extension. Moreover, the root fields of $f_{w,s}$ fulfill the
assumptions of Lemma 3.1, and there are infinitely many distinct quadratic
fields among the fields $\mathbb{Q}(\sqrt{\Delta_{s}})$.

If furthermore

  * b’)

$|w|\in[0.645,0.7348]$

then among those quadratic fields, there are infinitely many real ones.

A few remarks on the above theorem are in order.

######  Remark 3.

  * a)

There do indeed exist values $w\in\mathbb{Q}$ fulfilling all the assumptions
(including condition b’)) of the theorem, the easiest being $w=-2/3$, for
which the expression in a) is equivalent (up to squares) to $43$. In fact,
there are infinitely many such $w$; for example, the curve
$43Y^{2}=(-50W^{2}+27)(10W^{2}+8W+1)$ is a genus-$1$ curve with a rational
point (e.g., at $W=-2/3$), hence elliptic, and its rank is positive (which may
be quickly concluded from Mazur’s torsion theorem by simply enumerating more
than $16$ points; e.g., there are $20$ such points whose $W$-value has
numerator and denominator of absolute value $\leq 300$. Direct computation
with Magma actually yields that the rank is $2$). But rational points of an
elliptic curve of positive rank are well-known (by the “Poincaré-Hurwitz
theorem” [5, Satz 13]) to be locally dense in the real topology, whence
infinitely many values $w$ fulfilling all the conditions exist in the vicinity
of $w_{0}=-2/3$.

  * b)

A different, but related question is whether one may even find quadratic
fields $\mathbb{Q}(\sqrt{\Delta_{s}})$ as in the assertion such that any
prescribed finite set of primes is unramified in
$\mathbb{Q}(\sqrt{\Delta_{s}})$. This would require finding not just
infinitely many admissible values $w$ as above, but infinitely many pairwise
coprime squarefreee integers $D>0$ with all prime divisors congruent to $1$
modulo $3$, and such that the curve $DY^{2}=(-50W^{2}+27)(10W^{2}+8W+1)$ has
positive rank (since, indeed, calculation of $\Delta_{s}$ shows that the prime
divisors of $D$ ramify in $\mathbb{Q}(\sqrt{\Delta_{s}})$). Commonly believed
conjectures about distribution of ranks of twists of elliptic curves would
suggest that this can be achieved, and one may easily enumerate a great lot of
such values $D$, although proving an infinity result might be hard (and, in
fact. it is exactly the difficulty of such problems on polynomial values with
all prime divisors in some prescribed set which makes direct approaches to
Theorem 1.1 problematic). Nevertheless, even calculation of a few values $w$
(together with the corresponding discriminant $\Delta_{s}$) already shows
that, e.g., for any single prime $p$, there exist such fields
$\mathbb{Q}(\sqrt{\Delta_{s}})$ which are additionally unramified at $p$.

We furthermore note that an infinite set of values $s$ and of the associated
$\Delta_{s}$ as above can be effectively computed for each $w$ (as the proof
will show). We content ourselves with the case $w=-2/3$ (and specialization
values leading to imaginary quadratic fields), for which the following
explicit version may be deduced.

######  Corollary 3.3.

For all integers $u$ coprime to $30$, the (imaginary) quadratic number field

$\mathbb{Q}(\sqrt{-43\cdot(2^{11}\cdot 3^{10}\cdot 43^{6}u^{12}-43^{3}\cdot
263\cdot 883\cdot u^{6}+108)})$

possesses an everywhere unramified $SL_{2}(5)$-extension.

###### Proof of Theorem 3.2.

Begin with the family $f(a,t,X)=X^{2}(X-1)^{3}-t(X-a)$.

This yields an $S_{5}$-extension $E/\mathbb{Q}(a)(t)$ ramified at $t=0$ (with
inertia group generator of cycle structure $(3.2)$), at $t\mapsto\infty$ (with
inertia group generated by a $4$-cycle) and at the two roots
$\lambda_{1},\lambda_{2}$ of an irreducible quadratic polynomial $g_{a}(t)$
(with inertia group generated by a transposition). One may verify that the
residue extensions in $E$ at $t\mapsto\lambda_{i}$ are $S_{3}$-extensions
(i.e., the decomposition group at each of these branch points is isomorphic to
$C_{2}\times S_{3}\leq S_{5}$).

Step 1: Shrinking of residue extensions via pullback: In view of our
observations at the beginning of this section, we would rather prefer those
decomposition groups to be contained inside $C_{2}\times C_{3}$. Following the
approach of [8], we hope to arrive at such a scenario via a suitable quadratic
pullback $t=t(s)=c\cdot s^{2}$ (for a nonzero $c$ to be determined). This will
not succeed generically, but can be translated into a very explicit system of
equations, solvable via standard Groebner basis methods, which then
parameterizes those rational values of $a$ (and the corresponding values of
$c$) for which the pullback does succeed at shrinking the decomposition group.
Concretely, the branch points of the “pulled-back” extension
$E(s)/\mathbb{Q}(a)(s)$ are $s=0,\infty$, and the roots of
$cs^{2}=\lambda_{i}$ ($i=1,2$)222Note that it is “generically” sufficient to
do all the following for $i=1$, since the same result for $i=2$ follows from
algebraic conjugacy of the two branch points. We will verify a while later
that this algebraic conjugacy will in fact be preserved for all rational
values of $a$ relevant to us..

On the other hand, it is easy (using Magma) to find the discriminant
$\Delta_{i}\in\mathbb{Q}(a,\lambda_{i})$ of the residue extension at
$t\mapsto\lambda_{i}$ in $E/\mathbb{Q}(a)(t)$; the $2$-part of this residue
extension is then of course simply
$\mathbb{Q}(a,\lambda_{i})(\sqrt{\Delta_{i}})/\mathbb{Q}(a,\lambda_{i})$, and
since $\mathbb{Q}(a)(s)/\mathbb{Q}(a)(t)$ is unramified at
$t\mapsto\lambda_{i}$, the $2$-part of the residue extension at
$t\mapsto\lambda_{i}$ in the whole compositum $E(s)/\mathbb{Q}(a)(t)$ is the
compositum of quadratic extensions
$\mathbb{Q}(a,\lambda_{i})(\sqrt{\Delta_{i}},\sqrt{c\lambda_{i}})/\mathbb{Q}(a,\lambda_{i})$.
In particular, if
$\mathbb{Q}(a,\lambda_{i})(\sqrt{\Delta_{i}})=\mathbb{Q}(a,\sqrt{c\lambda_{i}})$
– which of course translates to $c\lambda_{i}\Delta_{i}$ being a non-zero
square in $\mathbb{Q}(a,\lambda_{i})$ – then the residue extensions at the
branch points $s\mapsto\pm\sqrt{\lambda_{i}/c}$ of $E(s)/\mathbb{Q}(a)(s)$ are
of odd degree. Since $\mathbb{Q}(a,\lambda_{i})=\mathbb{Q}(\sqrt{p(a)})$ for a
suitable polynomial $p(a)\in\mathbb{Z}(a)$, we can explicitly write
$c\lambda_{i}\Delta_{i}=\alpha_{1}+\alpha_{2}\sqrt{p(a)}$ (with
$\alpha_{i}\in\mathbb{Q}(a)$), and equate coefficients (at $1$ and
$\sqrt{p(a)}$) in the equation
$(\alpha_{1}+\alpha_{2}\sqrt{p(a)})=(u+v\sqrt{p(a)})^{2}$, yielding a
reasonably managable system of equations in the unknowns $a,c,u,v$ (indeed,
since $c$ is unique only up to squares, we may even demand $u=1$). It turns
out that solutions of this system are parameterized by a (rational) curve of
genus $0$, and it is then standard (again, when assisted by a computer) to
find a parameter $w$ of this curve. We are therefore able to express $a$ and
$c$ as rational functions in $\mathbb{Q}(w)$. In our case, we concretely
obtained $a=2w^{2}$, and $c=2\cdot(27-50w^{2})(10w^{2}+8w+1)$, which yields
exactly the polynomial $f_{w,s}(X)$ in the assertion of the theorem.

We may now plug in rational values for $w\in\mathbb{Q}$ in the polynomial
$f_{w}(s,X)$. Apart from finitely many values $w$,333Concretely, an exception
occurs only for values $w$ lying on the boundary of the underlying “moduli
space”, i.e., which make two or more branch points of the extension coalesce.
Calculation of the discriminant shows quickly that, in fact, this does not
happen for any rational $w\neq 0$. this parameterizes a $\mathbb{Q}$-regular
$S_{5}$-extension, say $F^{w}/\mathbb{Q}(s)$. We can also verify now that the
two non-zero finite branch points of the extension given by
$X^{2}(X-1)^{3}-t(X-2w^{2})=0$ are in fact always algebraically conjugate (a
generically fulfilled assumption which we used in the calculations above):
indeed, straightforward calculation of discriminants shows that $w$-values for
which both these points become rational would have to correspond to rational
points on the curve $Y^{2}=(10W^{2}+8W+1)(10W^{2}-8W+1)$; this is isomorphic
to the elliptic curve $Y^{2}=X(X-15)(X-24)$, and Magma confirms that this
elliptic curve has rank $0$ and no rational points apart from the $2$-torsion
- these are the points corresponding to $w=0$ and $w=\infty$, which are
already excluded anyways.

Step 2: Local behavior of specializations (one prime at a time): So far, we
have worked on the geometric level, and basically (up to killing ramification
at $0$ and $\infty$, which is easily achieved) obtained extensions fulfilling
the assumptions of Lemma 3.1 over $\mathbb{Q}(s)$ instead of $\mathbb{Q}$. The
tool to pull these results back to $\mathbb{Q}$ is Theorem 2.2, but we face
the problem that there are always finitely many “exceptional” primes. We will
now make some additional considerations on residue extensions of the extension
$F^{w}/\mathbb{Q}(s)$, in order to control the exceptional primes in Theorem
2.2. Firstly, the decomposition group at $t\mapsto 0$ of the splitting field
of $g(t,X):=X^{2}(X-1)^{3}-t(X-a(w))$ over $\mathbb{Q}(t)$ is the split
extension $D_{6}=C_{6}\rtimes C_{2}$ of the inertia group
$\langle(1,2)(3,4,5)\rangle\cong C_{6}$. The completion of the above splitting
field over $t\mapsto 0$ therefore contains a totally ramified degree-$6$
subextension, and it is well known that such an extension is necessarily of
the form $\mathbb{Q}((\sqrt[6]{bt}))/\mathbb{Q}((t))$ for some
$b\in\mathbb{Q}^{\times}$, see, e.g., [12, pp.52–53]. Furthermore, since the
residue extension at a branch point of ramification index $e$ always contains
the $e$-th roots of unity, this residue extension must equal
$\mathbb{Q}(\zeta_{3})$. The completion at $t\mapsto 0$ therefore equals
$\mathbb{Q}(\zeta_{3})((\sqrt[6]{bt}))$ for some $b\in\mathbb{Q}^{\times}$. To
determine the value of $b$, one first lets $X\to 0$ and then $Y:=X-1\to 0$, to
see that the completion contains the fields $\mathbb{Q}((\sqrt{at}))$ as well
as $\mathbb{Q}((\sqrt[3]{(1-a)t}))$, making the completion in total equal to
$\mathbb{Q}(\zeta_{3})((\sqrt[6]{\frac{a^{3}}{(1-a)^{2}}t}))$.

Now let the transcendental $r$ fulfill $r^{6}=c(w)\cdot\gamma^{2}t$ (with
$\gamma\neq 0$ to be determined). The splitting field of
$g(t(r),X)=X^{2}(X-1)^{3}-\frac{r^{6}}{c\gamma^{2}}(X-a)=X^{2}(X-1)^{3}-c(\frac{r^{3}}{c\gamma})^{2}(X-a)$
over $\mathbb{Q}(r)$ is then clearly also a pullback of $F^{w}/\mathbb{Q}(s)$;
it is furthermore unramified at $r\mapsto 0$ due to Abhyankar’s lemma (since
$\mathbb{Q}(r)/\mathbb{Q}(t)$ is totally ramified of ramification index $6$ at
$t\mapsto 0$). Combining with Krasner’s lemma (namely, specializing
$s=s(r)\in\mathbb{Q}(r)$ at values $r\mapsto r_{0}$ sufficiently close to $0$
in the $p$-adic topology), we can then conclude that every completion (at some
prime $p$) of $(F^{w}(r))_{r\mapsto 0}/\mathbb{Q}$ is also the completion of
some specialization $F^{w}_{s\mapsto s_{0}}/\mathbb{Q}$ at some non-branch-
point $s_{0}\in\mathbb{Q}$ of $F^{w}/\mathbb{Q}(s)$. To get hold of what this
completion looks like, we next identify $\textrm{Gal}((F^{w}(r))_{r\mapsto
0}/\mathbb{Q})$ as a subgroup of $\textrm{Gal}(F^{w}(r)/\mathbb{Q}(r))\cong
S_{5}$. The completion at $t\mapsto 0$ of the whole Galois closure of
$F^{w}(r)/\mathbb{Q}(t)$ is the compositum of
$\mathbb{Q}(\zeta_{3})((\sqrt[6]{\frac{a^{3}}{(1-a)^{2}}t}))=\mathbb{Q}(\zeta_{3})((\sqrt[6]{\frac{a^{3}}{c\gamma^{2}(1-a)^{2}}}\cdot
r))$ and $\mathbb{Q}((r))$, and hence equal to
$\mathbb{Q}(\zeta_{3},\sqrt[6]{\frac{a^{3}}{c\gamma^{2}(1-a)^{2}}})((r))$.
Setting now $\gamma:=(1-a)^{2}c$ and noting that $2a=4w^{2}$ is a square, we
see that the residue extension of $F^{w}(r)/\mathbb{Q}(r)$ at (the non-branch
point) $r\mapsto 0$ becomes the (at most) biquadratic extension
$\mathbb{Q}(\zeta_{3},\sqrt{\frac{c}{2}})=\mathbb{Q}(\zeta_{3},\sqrt{(-50w^{2}+27)(10w^{2}+8w+1)})$.
Note that any prime ramifying in this extension is either a divisor (of odd
multiplicity) of $c/2$, or equal to one of $2$, $3$ or $\infty$. Redoing the
above calculation individually for the completions of the (degree-$5$)
extension $\mathbb{Q}(x)/\mathbb{Q}(t)$ at the two places extending $t\mapsto
0$ (namely, $x\mapsto 0$ and $x\mapsto 1$), one furthermore gets an embedding
of the decomposition group $\textrm{Gal}((F^{w}(r))_{r\mapsto
0}/\mathbb{Q})\cong C_{2}\times C_{2}$ into $S_{5}$ identifying
$\textrm{Gal}(\mathbb{Q}(\sqrt{c/2})/\mathbb{Q})$ with $\langle(1,2)\rangle$
and $\textrm{Gal}(\mathbb{Q}(\zeta_{3})/\mathbb{Q})$ with
$\langle(3,4)\rangle$ (up to conjugation in $S_{5}$).

Assume now, as we have done in condition a) of the theorem, that we can find
$w$ such that all primes $p$ of odd multiplicity in $c/2$ are congruent to $1$
mod $3$. Then $p$ splits in $\mathbb{Q}(\zeta_{3})$ and conversely
$\sqrt{|c/2|}$ is in $\mathbb{Q}_{3}$. But note that condition b) on the
values $w$ ensures that $c>0$, whence $3$ is indeed split in
$\mathbb{Q}(\sqrt{c/2})$. We have therefore obtained that all the finite
ramified primes of $F^{w}(r)_{r\mapsto 0}/\mathbb{Q}$ dividing $c/2$, and also
the prime $3$, have decomposition group generated by a transposition in this
specialization.

Moreover, there exist values $s_{0}$ such that $F^{w}_{s_{0}}/\mathbb{Q}$ is
unramified at $2$; indeed, one verifies computationally (see the ancillary
file) that, for $w_{1}$ odd and $w_{2}$ even (resp, $w_{1}$ even and $w_{2}$
odd), the discriminant of $F^{w}_{s\mapsto(w_{2}^{3})/4}/\mathbb{Q}$ (resp.,
of $F^{w}_{s\mapsto 2w_{2}^{3}}$) is equal to the product of a square and an
integer congruent to $1$ modulo $4$; finally, when $w_{1}$ and $w_{2}$ are
both odd, the same holds for one of the two specializations $F^{w}_{s\mapsto
2w_{2}^{3}}/\mathbb{Q}$ and $F^{w}_{s\mapsto 6w_{2}^{3}}/\mathbb{Q}$, i.e., in
all cases, we have identified a specialization $F^{w}_{s_{0}}/\mathbb{Q}$ in
whose quadratic subextension the prime $2$ is unramified.

On the other hand, when $w_{2}$ is odd and $s_{0}=2w_{2}^{3}$ (and same for
$s_{0}=6w_{2}^{3}$), the defining polynomial $f_{w,s_{0}}(X)$ factors mod $2$
after suitable linear shifts as
$\frac{1}{8}f_{w_{1}/w_{2},2w_{2}^{3}}(1-2X)=(2X-1)^{2}\cdot(-X)^{3}+(50w_{1}^{2}-27w_{2}^{2})(10w_{1}^{2}+8w_{1}w_{2}+w_{2}^{2})(w_{2}^{2}(1-2X)-2w_{1}^{2})\equiv
X^{3}+1$ mod $2$, which is a separable cubic. This means that over
$\mathbb{Q}_{2}$ this polynomial has exactly three roots of non-negative
valuation, all in the unramified extension. Hence the inertia group at $2$ in
this specialization is at most cyclic of order $2$, generated by a
transposition (and the same result follows analogously for the other
specializations considered in the previous paragraph). But since we have also
just seen that this inertia group must be contained in $A_{5}$, we conclude
altogether that $2$ is unramified in this specialization. The case “$w_{2}$
even” is easier, since here the specialization
$f_{w_{1}/w_{2},w_{2}^{3}/4}(X)$ is congruent, up to constant, to a separable
quintic mod $2$.

Finally, the condition b’) ensures the existence of values $s_{0}$ such that
complex conjugation in $F^{w}_{s_{0}}/\mathbb{Q}$ is represented by the
identity, i.e., the extension is totally real. To verify this claim, one need
not try out infinitely many values $w$; rather, for a given $w$, the cycle
type of conjugation is locally constant as $s_{0}$ moves through
$\mathbb{P}^{1}(\mathbb{R})$ minus the set of branch points of $F^{w}$; and
furthermore, it is computable in a purely group-theoretical way in terms of
the “branch cycle description” of the cover corresponding to
$F^{w}/\mathbb{Q}(s)$ (see, e.g., [14, Chapter I, Theorem 10.3] for a precise
formula). This branch cycle description is again locally invariant and can
change only when $w$ passes through a boundary point of the “Hurwitz space”
underlying our family of extensions, i.e., when $w$ is specialized such that
two or more branch points collide. Via a calculation of discriminants, the
problem thus reduces to evaluating a finite number of $w$ and, for each
$w$-values, a finite number of $s$-values, counting the number of real roots
of $f_{w,s}(X)$ for each choice.

Step 3: Local behavior of specializations (all primes simultaneously): We are
now ready to invoke Theorem 2.2. Let $\mathcal{S}_{0}$ be the exceptional set
(as in Theorem 2.2) of finite primes for the extension $F^{w}/\mathbb{Q}(s)$.
We know already that, for each $p\in\mathcal{S}_{0}$, there exists a non-
branch-point $s(p)\in\mathbb{Q}$ such that $F^{w}_{s(p)}/\mathbb{Q}$ is either
unramified at $p$, or tamely ramified with cyclic decomposition group
generated by a transposition.444In fact, we have seen that we can avoid
ramification for all $p\in\mathcal{S}_{0}$, except possibly $3$, $\infty$ and
the primes of odd multiplicity in $c/2$. Combining with Krasner’s lemma, we
obtain the existence of a non-empty $\mathcal{S}_{0}$-adically open set (i.e.,
intersection of $p$-adic neighborhoods for $p\in\mathcal{S}_{0}$) of values
$s_{0}$ such that $F^{w}_{s_{0}}/\mathbb{Q}$ fulfills the above for all primes
$p\in\mathcal{S}_{0}$ simultaneously. We may even assume additionally that
$s_{0}$ is of the form $s_{0}=u^{3}/v^{2}$ with $u,v\in\mathbb{Z}$ coprime,
since the set of these numbers is easily seen to be dense in every
$\mathbb{Q}_{p}$.555This last assertion is true also for $p=\infty$, whence,
under the extra assumption of condition b’), we may also assume the
specialization $F^{w}_{s_{0}}$ to be totally real up to suitable choice of
$s_{0}$. This extra assumption has the effect that, when applying Theorem 2.1,
primes outside of $\mathcal{S}_{0}$ at which $s_{0}$ meets either the branch
point $s\mapsto 0$ or the branch point $s\mapsto\infty$ will remain unramified
in $F^{w}_{s_{0}}/\mathbb{Q}$ (since the ramification indices at these two
branch points are $3$ and $2$ respectively). Ramified primes, other than
primes in $\mathcal{S}_{0}$, will therefore all have inertia group generated
by a transposition, corresponding to the inertia group at any finite non-zero
branch point of $F^{w}/\mathbb{Q}(s)$. But at those branch points, by the
construction in Step 1, the decomposition group is contained in $C_{2}\times
C_{3}$; due to Theorem 2.2, the same will therefore hold for all ramified
primes (outside of $\mathcal{S}_{0}$) of $F^{w}_{s_{0}}/\mathbb{Q}$. Due to
the well-known compatibility of Hilbert’s irreducibility theorem with weak
approximation, we may even assume $F^{w}_{s_{0}}/\mathbb{Q}$ to have full
Galois group $S_{5}$ for infinitely many such $s_{0}$. It is also immediately
obvious that there are infinitely many distinct quadratic subextensions inside
these infinitely many specializations (for example, suitable choice of $s_{0}$
yields arbitrarily large primes with “transposition inertia”).

Step 4: Solving the embedding problem: In total, we have obtained infinitely
many (tamely ramified) specializations $F^{w}_{s_{0}}/\mathbb{Q}$ in which all
non-trivial inertia groups are generated by transpositions, and all
decomposition groups at such primes are cyclic (of order $2$ or $6$). By Lemma
3.1, each such specialization embeds into a quadratic extension $L\supseteq
F^{w}_{s_{0}}$ such that, with $k:=\mathbb{Q}(\sqrt{\Delta(F^{w}_{s_{0}})})$,
the extension $L/k$ is Galois of group $SL_{2}(5)$ and unramified at all
finite primes. When $k$ is imaginary, we are already done. Otherwise, we may
need to change $L$ slightly to make $L/k$ unramified also at the infinite
primes. Note that in this case, complex conjugation in $F^{w}_{s_{0}}/k$ would
have to be represented by an element of $A_{5}$, i.e., by the identity (the
case of a double transposition as complex conjugation does not occur, since
this would be a local obstruction to the embedding problem, which we have
already excluded).

So we are left with the case that $F^{w}_{s_{0}}$ is totally real, but $L$ is
not any more. In this case, we can twist $L/k$ to obtain a totally real (i.e.,
unramified at the archimedean prime) extension without changing ramification
at other primes. Recall that, from our considerations in Step 2, we may assume
that $s_{0}$ is chosen such that all primes dividing
$(-50w_{1}^{2}+27w_{2}^{2})(10w_{1}^{2}+8w_{1}w_{2}+w_{2}^{2})$ an odd number
of times have transposition inertia in $F^{w}_{s_{0}}/\mathbb{Q}$, and are
therefore in particular ramified in $k/\mathbb{Q}$. But note that, since at
least one of $w_{1},w_{2}$ is odd, the above product always evaluates either
to an integer congruent to $3$ mod $4$ or to four times such an integer. Also,
in the intervals allowing totally real specializations, both quadratic factors
take only positive values. This means that there is necessarily at least one
prime $p\equiv 3$ mod $4$ ramifying in $k/\mathbb{Q}$. We may then take take
the compositum $\widehat{L}$ of $L$ and $k(\sqrt{-p})$. This is an
$SL_{2}(5)\times C_{2}$-extension of $k$, and if $D\cong C_{2}$ denotes the
diagonal subgroup of the center of the group $SL_{2}(5)\times C_{2}$, then
$\tilde{L}^{D}/k$ is again an $SL_{2}(5)$-extension. Since $k/\mathbb{Q}$ is
already ramified at $p$, the extension $k(\sqrt{-p})/k$ is ramified exactly at
the archimedean primes. Hence, all non-archimedean primes still remain
unramified in $\tilde{L}^{D}/k$, whereas for the archimedean ones,666Note that
both archimedean primes of $k$ must behave identically in $L$, since
$L/\mathbb{Q}$ is Galois. the fact that $L/k$ and $k(\sqrt{-p})/k$ have the
same completion (namely $\mathbb{C}$) means that
$\tilde{L}^{D}\subset\mathbb{R}$. This completes the proof. ∎

Note once again that, using the above proof, Theorem 3.2 can be made fully
effective in the sense that, for each $w$ fulfilling the assumptions, an
explicit infinite set of values $s_{0}$ (and hence, via calculation of
discriminants, an explicit set of quadratic fields fulfilling the assertion)
can be computed; indeed, as we have seen, all $s_{0}=\frac{u^{3}}{v^{2}}$ with
$u,v\in\mathbb{Z}$ in some suitable intersection of finitely many $p$-adic
neighborhoods are admissible. Making these neighborhoods explicit is also
possible, although the result might not look too attractive.

For the proof of the explicit assertion of Corollary 3.3, it suffices to
investigate the local behavior of the specializations
$F^{w}_{s_{0}}/\mathbb{Q}$ (with $w=-2/3$) at the “exceptional” primes
somewhat more closely.

###### Proof of Corollary 3.3.

Setting $w=-2/3$ in the notation of the proof of Theorem 3.2, we have
$a(w)=8/9$ and $c(w)=86/81$. Up to multiplying $s$ with a suitable constant,
we thus have the defining polynomial $f(s,X)=X^{2}(X-1)^{3}-86s^{2}(9X-8)$ for
the extension $F^{w}/\mathbb{Q}(s)$. Using Remark 1, a calculation of
discriminants shows that the set of exceptional primes (in the sense of
Theorem 2.2) is here $\mathcal{S}_{0}\cup\\{\infty\\}$ with
$\mathcal{S}_{0}=\\{2,3,5,43,97\\}$. We want to specialize $s\mapsto
s_{0}=86u^{3}$ with $u\in\mathbb{Z}$ coprime to $30$, which (again, by a
discriminant calculation) gives exactly the quadratic subfields as in the
assertion of the corollary. Just as in the proof of Theorem 3.2, this
specialization “kills” ramification inherited from branch points with non-
transposition inertia, and we conclude that $F^{w}_{s_{0}}/\mathbb{Q}$ will
have no local obstruction for the central embedding problem at finite primes
outside of $\mathcal{S}_{0}$. We are therefore reduced to an investigation of
local behavior at those finitely many exceptional primes.

Regarding the prime $43$, note that if $s_{0}=86(43^{d}u^{\prime})^{3}$
($u^{\prime}\in\mathbb{Z}$ coprime to $43$, and $d\geq 0$), then the mod-$43$
reduction of the polynomial $\frac{1}{43^{6d+3}}f(s_{0},1-43^{2d+1}X)$ is the
separable cubic polynomial $-X^{3}+-2^{3}u^{\prime 6}$ with three roots. This
means that ${\textrm{Gal}}(f(s_{0},X)/\mathbb{Q}_{43})$ is either trivial or
generated by a transposition (in fact, since $43$ divides the discriminant of
$F^{w}_{s_{0}}$, the latter is the case), meaning that for these values
$s_{0}$, there is no local obstruction at $43$. The same reasoning works for
the primes $2$, $3$, $5$ and $97$ (in fact, one may even additionally deduce
that these primes are always unramified under the given conditions, as the
explicit expression for the discriminant of the quadratic subfield shows).

It thus remains to verify that $\textrm{Gal}(F^{w}_{s_{0}}/\mathbb{Q})$ is
always all of $S_{5}$. With our specifications, $f(s_{0},X)$ is always
separable modulo $5$, and either irreducible or divisible by an irreducible
degree-$4$ factor; similarly, $f$ is always separable modulo $3$, with
irreducible factors of degree $2$ and $3$.

Now $\textrm{Gal}(F^{w}_{s_{0}}/\mathbb{Q})=S_{5}$ follows from standard
results, since a subgroup of $S_{5}$ containing an element of cycle structure
$(3.2)$ and one of cycle structure either $(5)$ or $(4.1)$ is necessarily all
of $S_{5}$. ∎

##  Appendix A Remarks on the case $G=SL_{2}(9)$

We conclude by noting that the above approach could in principle work
analogously to solve Conjecture 1 also for the group $G=SL_{2}(9)$ In
particular, due to the isomorphism $PSL_{2}(9)\cong A_{6}$, one may proceed in
analogy to the previous section, by first producing Galois extensions of
$\mathbb{Q}$ with Galois group $S_{6}$ and suitable local behavior, and then
embedding them into extensions with group $2.S_{6}$, the degree-$2$ central
extension of $S_{6}$ in which the transpositions are split. The latter group
has an index-$2$ normal subgroup isomorphic to $SL_{2}(9)$. We thus have a
sufficient criterion for the solvability of the embedding problem similar to
Lemma 3.1:

######  Lemma A.1.

Let $\Omega/\mathbb{Q}$ be a Galois extension with group $S_{6}$ such that

  * 1)

all inertia groups at ramified primes of $\Omega/\mathbb{Q}$ are generated by
transpositions, and

  * 2)

all decomposition groups at ramified primes are conjugate in $S_{6}$ to one of
$\langle(1,2)\rangle$, $\langle(1,2),(3,4)(5,6)\rangle$ and
$\langle(1,2),(3,4,5)\rangle$.

Then, if $k\subset\Omega$ denotes the fixed field of $A_{6}$, the extension
$\Omega/k$ embeds into an $SL_{2}(9)$-extension unramified at all finite
primes.

###### Proof.

It suffices to note that the first and third group in condition 2) are split
in the extension $2.S_{6}$ (and thus pose no local obstruction to the central
embedding problem), whereas the second one lifts to an abelian group
isomorphic to $C_{2}\times C_{4}$ (with the subgroup generated by the
transposition still split); in case a prime $p$ has this decomposition group
(and transposition inertia) in $\Omega/\mathbb{Q}$, there is no local
obstruction to the embedding problem either, since the local extension is then
isomorphic to the compositum of a totally ramified quadratic extension with
the unramified quadratic extension, and the local embedding problem is solved
by extending the quadratic unramified extension to the degree-$4$ one. The
rest of the proof is simply application of Propositions 2.3 and 2.4, as for
Lemma 3.1. ∎

In order to enforce conditions 1) and 2) of the above lemma, one may again
construct geometric Galois realizations with inertia groups generated by
transpositions, and this time such that the decomposition group at such branch
points is contained in
$\textrm{Sym}(\\{1,2\\})\times\textrm{Alt}(\\{3,4,5,6\\})$; this is again a
subgroup of index $2$ in the largest possible decomposition group $S_{2}\times
S_{4}$ for a “transposition branch point”, and hence enforcing such a
situation corresponds to a process of “swallowing” discriminants just as in
the previous section, which may again be performed using certain $4$-branch
point families. So far, however, I have not succeeded in additionally
controlling the “small” (i.e., exceptional) primes for such families, and it
may well be necessary to increase the branch point number further to achieve
this.

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