# Galois correspondence for Galois coverings on tropical curves

Song JuAe  Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo,
192-0397, Japan<EMAIL_ADDRESS>
###### Abstract.

We define Galois coverings on tropical curves for which a Galois
correspondence and a universal mapping property hold.

###### Key words and phrases:

Galois correspondence for Galois coverings on tropical curves, universal
mapping property

###### 2020 Mathematics Subject Classification:

Primary 14T15; Secondary 14T20

##  1\. Introduction

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a map between tropical curves
and $G$ a finite group. For an isometric action of $G$ on $\varGamma$,
$\varphi$ is Galois if (1) $\varphi$ is a finite harmonic morphism, (2) the
order of $G$ coincides with the degree of $\varphi$, (3) the action of $G$ on
$\varGamma$ induces a transitive action on every fiber, and (4) every
stabilizer subgroup of $G$ with respect to all but a finite number of points
is trivial. We also say $\varphi$ to be $G$-Galois. Here, a tropical curve is
a metric graph that may have edges of length $\infty$, and a finite harmonic
morphism is defined as a morphism of our category of tropical curves (see
Section 2 for more details). When $\varphi$ is $G$-Galois, for a finite
harmonic morphism $\psi:\varGamma\to\varGamma^{\prime\prime}$, we define
$G(\psi)$ as the subset of $G$ consisting of all elements $g$ of $G$
satisfying $\psi\circ g=\psi$. For a fixed $G$-Galois covering
$\varphi:\varGamma\to\varGamma^{\prime}$, let $A^{\prime}$ be the set of all
finite haromonic morphisms
$\widetilde{\psi}:\varGamma\to\widetilde{\varGamma^{\prime\prime}}$ each of
which is $G(\widetilde{\psi})$-Galois. For two elements $\psi_{1}$ and
$\psi_{2}$ of $A^{\prime}$, we write $\psi_{1}\sim\psi_{2}$ if there exists a
finite harmonic morphism of degree one $\psi_{12}$ satisfying
$\psi_{1}=\psi_{12}\circ\psi_{2}$. This $\sim$ becomes an equivalence
relation. Let $A:=A^{\prime}/\\!\\!\sim$. For $[\psi_{i}]\in A$, where
$[\psi_{i}]$ denotes the equivalence class of $\psi_{i}$, we write
$[\psi_{1}]\leq_{A}[\psi_{2}]$ if for any $\psi^{\prime}_{1}\in[\psi_{1}]$ and
$\psi^{\prime}_{2}\in[\psi_{2}]$, there exists a finite harmonic morphism
$\theta$ satisfying $\psi^{\prime}_{1}=\theta\circ\psi^{\prime}_{2}$. This
$\leq_{A}$ is well-defined and becomes a partial order. Let $B$ be the set of
all subgroups of $G$. For $G_{i}\in B$, we define a partial order $\leq_{B}$
such that $G_{1}\leq_{B}G_{2}$ if $G_{1}\subset G_{2}$. Then, we have the
following:

######  Theorem 1.1 (Galois correspondence for Galois coverings).

In the above setting, there exists a one-to-one correspondence between
$(A,\leq_{A})$ and $(B,\leq_{B})$ reversing (partial) orders.

For Galois coverings, the following two theorems hold. Here, for a map between
tropical curves $\varphi:\varGamma\to\varGamma^{\prime}$ and an isometric
action of a finite group $H$ on $\varGamma$, $\varphi$ is $H$-normal if
$\varphi$ satisfies the conditions (1), (3), and (4) above.

######  Theorem 1.2.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a $G$-Galois covering,
$\psi:\varGamma\to\varGamma^{\prime\prime}$ in $A^{\prime}$, and
$\theta:\varGamma^{\prime\prime}\to\varGamma^{\prime}$ the unique continuous
map satisfying $\varphi=\theta\circ\psi$. Then, the following are equivelent:

$(1)$ $G(\psi)$ is a normal subgroup of $G$,

$(2)$ there exists a finite group $H$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H$-Galois, and

$(3)$ there exists a finite group $H^{\prime}$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H^{\prime}$-normal.

Moreover, suppose in addition to $(2)$ that for any $g\in G$, there exists
$h\in H$ satisfying $\psi\circ g=h\circ\psi$, then $H$ is isomorphic to the
quotient group $G/G(\psi)$.

######  Theorem 1.3 (Universal mapping property).

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be $G$-Galois. Then, for any
finite harmonic morphism $\psi:\varGamma\to\varGamma^{\prime\prime}$
satisfying for any $g\in G$, $\psi\circ g=\psi$, there exists a unique finite
harmonic morphism $\theta$ satisfying $\psi=\theta\circ\varphi$.

Our definition of Galois coverings on tropical curves is an enhanced version
of the definition in [2] (the condition (4) is added). By adding the condition
(4), Theorems 1.1 and 1.3 above and Theorem 1.4 below, which will be shown in
[3], hold; see Remark 3.7 and Examples 3.8, 3.9, 3.10.

######  Theorem 1.4 ([3]).

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a finite harmonic morphism
between tropical curves and $G$ a finite group isometrically acting on
$\varGamma$. Then, $\varphi$ is $G$-Galois if and only if the action of $G$ on
$\operatorname{Rat}(\varGamma)$ naturally induced by the action of $G$ on
$\varGamma$ is Galois for the semifield extension
$\operatorname{Rat}(\varGamma)/\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$.

Here, $\operatorname{Rat}(\varGamma)$ denotes the semifield of all rational
functions on $\varGamma$ and
$\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ the pull-back. In the
setting of Theorem 1.4, “the action of $G$ on $\operatorname{Rat}(\varGamma)$
is Galois for
$\operatorname{Rat}(\varGamma)/\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$”
means that the $G$-invariant subsemifield of $\operatorname{Rat}(\varGamma)$
is $\varphi^{\ast}(\operatorname{Rat}(\varGamma^{\prime}))$ and that subgroups
of $G$ whose invariant semifields coincide are equal.

This paper is organized as follows. In Section 2, we give the definitions of
tropical curves and finite harmonic morphisms between tropical curves. Section
3 gives proofs of Theorems 1.1, 1.2, and 1.3.

## Acknowledgements

The author thanks my supervisor Masanori Kobayashi, Yuki Kageyama, and
Yasuhito Nakajima for helpful comments. This work was supported by JSPS
KAKENHI Grant Number 20J11910.

##  2\. Preliminaries

In this section, we prepare basic definitions related to tropical curves which
we need later. See, for example, [1] for details.

In this paper, a graph is an unweighted, undirected, finite, connected
nonempty multigraph that may have loops. For a graph $G$, the set of vertices
is denoted by $V(G)$ and the set of edges by $E(G)$. The degree of a vertex is
the number of edges incident to it. Here, a loop is counted twice. A leaf end
is a vertex of degree one. A leaf edge is an edge incident to a leaf end. A
tropical curve is the underlying topological space of the pair $(G,l)$ of a
graph $G$ and a function $l:E(G)\to{\boldsymbol{R}}_{>0}\cup\\{\infty\\}$
called a length function, where $l$ can take the value $\infty$ on only leaf
edges, together with an identification of each edge $e$ of $G$ with the closed
interval $[0,l(e)]$. When $l(e)=\infty$, the interval $[0,\infty]$ is the one
point compactification of the interval $[0,\infty)$ and the leaf end of $e$
must be identified with $\infty$. We regard $[0,\infty]$ not just as a
topological space but as almost a metric space. The distance between $\infty$
and any other point is infinite. If $E(G)=\\{e\\}$ and $l(e)=\infty$, then we
can identify either leaf ends of $e$ with $\infty$. When a tropical curve
$\varGamma$ is obtained from $(G,l)$, the pair $(G,l)$ is called a model for
$\varGamma$. There are many possible models for $\varGamma$. A model $(G,l)$
is loopless if $G$ is loopless. Let $\varGamma_{\infty}$ denote the set of all
points of $\varGamma$ identified with $\infty$. An element of
$\varGamma_{\infty}$ is called a point at infinity. We frequently identify a
vertex (resp. an edge) of $G$ with the corresponding point (resp. the
corresponding closed subset) of $\varGamma$. The relative interior $e^{\circ}$
of an edge $e$ is $e\setminus\\{v,w\\}$ with the endpoint(s) $v,w$ of $e$.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a continuous map between
tropical curves. $\varphi$ is a finite morphism if there exist loopless models
$(G,l)$ and $(G^{\prime},l^{\prime})$ for $\varGamma$ and
$\varGamma^{\prime}$, respectively, such that $(1)$ $\varphi(V(G))\subset
V(G^{\prime})$ holds, $(2)$ $\varphi(E(G))\subset E(G^{\prime})$ holds, and
$(3)$ for any edge $e$ of $G$, there exists a positive integer
$\operatorname{deg}_{e}(\varphi)$ such that for any points $x,y$ of $e$,
$\operatorname{dist}(\varphi(x),\varphi(y))=\operatorname{deg}_{e}(\varphi)\cdot\operatorname{dist}(x,y)$
holds. Here, $\operatorname{dist}(x,y)$ denotes the distance between $x$ and
$y$. Assume that $\varphi$ is a finite morphism. $\varphi$ is harmonic if for
every vertex $v$ of $G$, the sum $\sum_{e\in E(G):\,e\mapsto e^{\prime},\,v\in
e}\operatorname{deg}_{e}(\varphi)$ is independent of the choice of
$e^{\prime}\in E(G^{\prime})$ incident to $\varphi(v)$. This sum is denoted by
$\operatorname{deg}_{v}(\varphi)$. Then, the sum $\Sigma_{v\in V(G):\,v\mapsto
v^{\prime}}\operatorname{deg}_{v}(\varphi)$ is independent of the choice of
vertex $v^{\prime}$ of $G^{\prime}$. It is said the degree of $\varphi$ and
written by $\operatorname{deg}(\varphi)$. If both $\varGamma$ and
$\varGamma^{\prime}$ are singletons, we regard $\varphi$ as a finite harmonic
morphism that can have any number as its degree. Note that if
$\varphi\circ\psi$ is a composition of finite harmonic morphisms, then it is
also a finite harmonic morphism of degree
$\operatorname{deg}(\varphi)\cdot\operatorname{deg}(\psi)$, and thus tropical
curves and finite harmonic morphisms between them make a category.

######  Remark 2.1.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a map between tropical curves.
Then $\varphi$ is a continuous map whose restriction on
$\varGamma\setminus\varGamma_{\infty}$ is an isometry if and only if it is a
finite harmonic morphism of degree one. In this paper, we will use the word “a
finite group $G$ isometrically acts on a tropical curve $\varGamma$” as the
meaning that $G$ continuously acts on $\varGamma$ and it is isometric on
$\varGamma\setminus\varGamma_{\infty}$.

######  Remark 2.2.

Let $\varGamma$ be a tropical curve and $G$ a finite group isometrically
acting on $\varGamma$. Let $\varGamma/G$ be the quotient space (as a
topological space) and $\pi_{G}:\varGamma\to\varGamma/G$ be the natural
surjection. Fix a loopless model $(V,E,l)$ ($V$ is a set of vertices and $E$
is a set of edges) for $\varGamma$ such that for any $g\in G$ and $e\in E$,
$g(V)=V$ holds and if $g(e)=e$, then the restriction of $g$ on $e$ is the
identity map of $e$. Note that by the proof of [1, Lemma 3.4], we can choose
such a loopless model. Let $V^{\prime}:=\pi_{G}(V)$, $E^{\prime}:=\pi_{G}(E)$
and for any $e\in E$, $l^{\prime}(\pi_{G}(e)):=|G_{e}|\cdot l(e)$, where
$G_{e}$ denotes the stabilizer subgroup of $G$ with respect to $e$ and
$|G_{e}|$ the order of $G_{e}$. Then, $(V^{\prime},E^{\prime},l^{\prime})$
gives $\varGamma/G$ a tropical curve structure and is a loopless model for the
quotient tropical curve $\varGamma/G$. By loopless models $(V,E,l)$ and
$(V^{\prime},E^{\prime},l^{\prime})$ for $\varGamma$ and $\varGamma/G$,
respectively, $\pi_{G}$ is a finite harmonic morphism of degree $|G|$.

##  3\. Main results

In this section, we give our definition of Galois coverings on tropical curves
and proofs of Theorems 1.1, 1.2, and 1.3.

######  Definition 3.1.

Let $\varGamma$ be a tropical curve and $G$ a finite group. An isometric
action of $G$ on $\varGamma$ is Galois if there exists a finite subset
$U^{\prime}$ of $\varGamma/G$ such that for any
$x^{\prime}\in(\varGamma/G)\setminus U^{\prime}$,
$|\pi^{-1}_{G}(x^{\prime})|=|G|$ holds.

######  Definition 3.2.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a map between tropical curves.
$\varphi$ is Galois if there exists a Galois action of a finite group $G$ on
$\varGamma$ such that there exists a finite haramonic morphism of degree one
$\theta:\varGamma/G\to\varGamma^{\prime}$ satisfying $\varphi\circ
g=\theta\circ\pi_{G}$ for any $g\in G$. Then, we say that $\varphi$ is a
$G$-Galois covering on $\varGamma^{\prime}$ or just $G$-Galois.

###### Proof of Theorem 1.1.

Let

$\displaystyle\Phi:A\to B;\qquad[\psi]\mapsto G(\psi)$

and

$\displaystyle\Psi:B\to A;\qquad G^{\prime}\mapsto[\pi_{G^{\prime}}].$

We shall show that if $\psi_{1}\sim\psi_{2}$ holds, then
$G(\psi_{1})=G(\psi_{2})$ holds. For any $f\in G(\psi_{2})$, we have

$\displaystyle\psi_{1}\circ f=(\psi_{12}\circ\psi_{2})\circ
f=\psi_{12}\circ(\psi_{2}\circ f)=\psi_{12}\circ\psi_{2}=\psi_{1}.$

Thus $f\in G(\psi_{1})$ holds. The inverse inclusion can be shown similarly.
Therefore, $\Phi$ is well-defined.

We shall show that for any $G^{\prime}\in B$, the action of $G^{\prime}$ on
$\varGamma$ induced by that of $G$ on $\varGamma$ is Galois. Let $U^{\prime}$
be the finite subset of $\varGamma/G$ in Definition 3.1. For any
$x\in\varGamma\setminus\pi^{-1}_{G}(U^{\prime})$, we have
$|G_{x}|=\frac{|G|}{|Gx|}=1$, where we use the orbit-stabilizer theorem (cf.
[4, Chapter 6]) and $Gx$ stands for the orbit of $x$ by $G$, and thus
$|G^{\prime}_{x}|=1$. Therefore, $\pi_{G^{\prime}}(\pi^{-1}_{G}(U^{\prime}))$
is finite and for any
$x^{\prime}\in(\varGamma/G^{\prime})\setminus\pi_{G^{\prime}}(\pi^{-1}_{G}(U^{\prime}))$,
we have $|\pi^{-1}_{G^{\prime}}(x^{\prime})|=|G^{\prime}|$.

We shall show that for any $G^{\prime}\in B$, $G^{\prime}=G(\pi_{G^{\prime}})$
holds. By the definition of $G(\pi_{G^{\prime}})$, $G^{\prime}\subset
G(\pi_{G^{\prime}})$ holds. Let $g\in G(\pi_{G^{\prime}})$. For any
$x\in\varGamma$, since $(\pi_{G^{\prime}}\circ g)(x)=\pi_{G^{\prime}}(x)$,
there exists $g^{\prime}\in G^{\prime}$ such that $g(x)=g^{\prime}(x)$. Let us
assume that $x$ is in $\varGamma\setminus\pi^{-1}_{G}(U^{\prime})$. Then
$g=g^{\prime}$ must hold. In fact, if $g\not=g^{\prime}$, then $g^{-1}g\not=1$
and $g^{-1}g^{\prime}(x)=x$. Thus, $g^{-1}g^{\prime}\in G_{x}\setminus\\{1\\}$
and it contradicts that $|\pi^{-1}_{G}(x)|=|G|$. By these arguments, we have
$[\pi_{G^{\prime}}]\in A$ and $\Phi\circ\Psi=\operatorname{id}_{B}$, where
$\operatorname{id}_{B}$ denotes the identity map of $B$.

We shall show that $\Psi\circ\Phi=\operatorname{id}_{A}$ holds. By the
definitions of $A$ and Galois coverings, for any $\psi\in A^{\prime}$,
$\psi\sim\pi_{G(\psi)}$ holds, and hence we have

$(\Psi\circ\Phi)([\psi])=[\pi_{G(\psi)}]=[\psi].$

We shall show that if $[\psi_{1}]\leq_{A}[\psi_{2}]$ holds, then
$G(\psi_{1})\geq_{B}G(\psi_{2})$ holds. For $g\in G(\psi_{2})$, we have
$\psi_{2}\circ g=\psi_{2}$. Since $[\psi_{1}]\leq_{A}[\psi_{2}]$, there exists
a finite harmonic morphism $\theta$ satisfying $\psi_{1}=\theta\circ\psi_{2}$.
Therefore

$\displaystyle\psi_{1}\circ g=(\theta\circ\psi_{2})\circ
g=\theta\circ(\psi_{2}\circ g)=\theta\circ\psi_{2}=\psi_{1}$

hold. Hence $g\in G(\psi_{1})$.

Finally, we shall show that if $G_{1}\leq_{B}G_{2}$ holds, then
$[\pi_{G_{1}}]\geq_{A}[\pi_{G_{2}}]$ holds. Since $G_{1}\subset G_{2}$, there
exists a unique continuous map $\theta$ satisfying
$\theta\circ\pi_{G_{1}}=\pi_{G_{2}}$. We shall show $\theta$ is a finite
harmonic morphism. Let $(V,E,l),(V_{1},E_{1},l_{1})$ and $(V_{2},E_{2},l_{2})$
be loopless models for $\varGamma$, $\varGamma/G_{1}$ and $\varGamma/G_{2}$,
respectively, compatible with $\pi_{G_{i}}$, i.e., $\pi_{G_{i}}(V)=V_{i}$ for
$i=1,2$. Let $e\in E$. Since each $\pi_{G_{i}}$ is Galois, for any points
$x,y$ of $e$, we have

$\displaystyle\operatorname{dist}(\pi_{G_{2}}(x),\pi_{G_{2}}(y))=\operatorname{dist}(x,y)=\operatorname{dist}(\pi_{G_{1}}(x),\pi_{G_{1}}(y)).$

Hence $\theta$ is a finite morphism and
$\operatorname{deg}_{\pi_{1}(e)}(\theta)=1$. Next, we shall show that $\theta$
is harmonic. Fix a vertex $v^{\prime}\in V_{1}$ and an edge
$e^{\prime\prime}\in E_{2}$ incident to $\theta(v^{\prime})$. Since the action
of $G$ on $\varGamma$ is isometric and $G_{1}\subset G_{2}$, with any
$v\in\pi^{-1}_{G_{2}}(\theta(v^{\prime}))$, we have

$\displaystyle\sum_{e^{\prime}\in E_{1}:\,e^{\prime}\mapsto
e^{\prime\prime},\,v^{\prime}\in
e^{\prime}}\operatorname{deg}_{e^{\prime}}(\theta)$
$\displaystyle=\frac{1}{|\theta^{-1}(\theta(v^{\prime}))|}\cdot\sum_{e^{\prime}\in
E_{1}:e^{\prime}\mapsto
e^{\prime\prime}}\operatorname{deg}_{e^{\prime}}(\theta)$
$\displaystyle=\frac{1}{|\theta^{-1}(\theta(v^{\prime}))|}\cdot|\\{e^{\prime}\in
E_{1}\,|\,\theta(e^{\prime})=e^{\prime\prime}\\}|$

and

$\displaystyle|\theta^{-1}(\theta(v^{\prime}))|=\frac{|\pi_{G_{2}}^{-1}(\theta(v^{\prime}))|}{|G_{1}v|}=\frac{|G_{2}v|}{|G_{1}v|}.$

With any edge $e\in E$ such that $\pi_{G_{2}}(e)=e^{\prime\prime}$, the ratio
$\frac{|G_{2}e|}{|G_{1}e|}=\frac{|G_{2}|}{|G_{1}|}$ coincides with the number
of elements of the inverse image of $e^{\prime\prime}$ by $\theta$. Therefore
we have

$\displaystyle\sum_{e^{\prime}\in E_{1}:\,e^{\prime}\mapsto
e^{\prime\prime},\,v^{\prime}\in
e^{\prime}}\operatorname{deg}_{e^{\prime}}(\theta)=\frac{|G_{1}v|}{|G_{2}v|}\cdot\frac{|G_{2}|}{|G_{1}|}=\frac{|{G_{2}}_{v}|}{|{G_{1}}_{v}|}\in\boldsymbol{Z}_{>0},$

where we use the orbit-stabilizer theorem at the second equality. Hence the
sum is independent of the choice of $e^{\prime\prime}$ and thus $\theta$ is
harmonic. ∎

Next, we will prove Theorem 1.2. For this purpose, we define normal coverings
on tropical curves:

######  Definition 3.3.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a map between tropical curves.
$\varphi$ is normal if there exists a Galois action of a finite group $H$ on
$\varGamma$ and a finite harmonic morphism
$\theta:\varGamma/H\to\varGamma^{\prime}$ such that for any $h\in H$,
$\varphi\circ h=\theta\circ\pi_{H}$ holds. Then, we say that $\varphi$ is a
$H$-normal covering on $\varGamma^{\prime}$ or just $H$-normal.

By definition, if $\varphi$ is $G$-Galois, then it is $G$-normal. Normal
coverings have the following property:

######  Proposition 3.4.

Let $\varphi:\varGamma\to\varGamma^{\prime}$ be a $H$-normal covering on
$\varGamma^{\prime}$. Then, for any finite harmonic morphism
$\psi:\varGamma^{\prime\prime}\rightarrow\varGamma$, there exists a loopless
model $(G^{\prime\prime},l^{\prime\prime})$ for $\varGamma^{\prime\prime}$
such that for any $e^{\prime\prime}\in E(G^{\prime\prime})$ and any
automorphim $f$ of $\varGamma^{\prime\prime}$, i.e., a finite harmonic
morphism of degree one $\varGamma^{\prime\prime}\to\varGamma^{\prime\prime}$,
satisfying $\varphi\circ\psi\circ f=\varphi\circ\psi$, there exists $h\in H$
such that $(\psi\circ
f)|_{e^{\prime\prime}}=(h\circ\psi)|_{e^{\prime\prime}}$, where
$|_{e^{\prime\prime}}$ denotes the restriction of the map on
$e^{\prime\prime}$.

###### Proof.

By the proof of [1, Lemma 3.4], we can choose
$(G^{\prime\prime},l^{\prime\prime})$ such that $\psi(V(G^{\prime\prime}))$
(resp. $(\varphi\circ\psi)(V(G^{\prime\prime}))$) induces a loopless model for
$\varGamma$ (resp. $\varGamma^{\prime}$) compatible with $\psi$ (resp.
$\varphi\circ\psi$) and for any $e^{\prime\prime}\in E(G^{\prime\prime})$ and
$h\in H$, if $h(\psi(e^{\prime\prime}))=\psi(e^{\prime\prime})$, then the
restriction $h|_{\psi(e^{\prime\prime})}$ is the identity map on
$\psi(e^{\prime\prime})$. For any edge $e^{\prime\prime}\in
E(G^{\prime\prime})$ and any point $x^{\prime\prime}$ of
$(e^{\prime\prime})^{\circ}$, since $(\varphi\circ\psi\circ
f)(x^{\prime\prime})=(\varphi\circ\psi)(x^{\prime\prime})$, there exists $h\in
H$ satisfying $(\psi\circ
f)(x^{\prime\prime})=(h\circ\psi)(x^{\prime\prime})$. Since the action of $H$
on $\varGamma$ is isometric and $h$ does not invert any edge, $(\psi\circ
f)|_{(e^{\prime\prime})^{\circ}}=(h\circ\psi)|_{(e^{\prime\prime})^{\circ}}$
holds. Since both $\psi\circ f$ and $h\circ\psi$ are continuous, $(\psi\circ
f)|_{e^{\prime\prime}}=(h\circ\psi)|_{e^{\prime\prime}}$ holds. ∎

This proposition is an analogue of the fact in the field theory that for an
algebraic extension $L/K$ of fields, it is normal if and only if all
embeddings $\sigma$ of $L$ to the algebraic closure of $K$ containing $L$
which is the identity on $K$ satisfy $\sigma(L)=L$. In the tropical curve
case, however, we must consider only an edge instead of the whole tropical
curve.

Let $\varphi:\varGamma\rightarrow\varGamma^{\prime}$ be a $G$-Galois covering
on $\varGamma^{\prime}$. For any
$\psi:\varGamma\rightarrow\varGamma^{\prime\prime}$ in $A^{\prime}$, since
$G(\psi)$ is a subgroup of $G$ and $\varGamma^{\prime}$ and
$\varGamma^{\prime\prime}$ have quotient topologies, there exists a unique
continuous map $\theta:\varGamma^{\prime\prime}\to\varGamma^{\prime}$ such
that $\varphi=\theta\circ\psi$.

######  Proposition 3.5.

In the above setting, the following are equivalent:

$(1)$ $G(\psi)$ is a normal subgroup of $G$,

$(2)$ there exists a finite group $H$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H$-Galois and for any $g\in
G$, there exists $h\in H$ satisfying $\psi\circ g=h\circ\psi$, and

$(3)$ there exists a finite group $H^{\prime}$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H^{\prime}$-normal and for
any $g\in G$, there exists $h^{\prime}\in H^{\prime}$ satisfying $\psi\circ
g=h^{\prime}\circ\psi$.

Moreover, if the above conditions hold, then $H$ in $(2)$ is isomorphic to the
quotient group $G/G(\psi)$.

For a map between tropical curves, we will use the word “it is a local
isometry” as the meaning that it is continuous on the whole domain tropical
curve and that it is a local isometry on the domain tropical curve except all
points at infinity.

###### Proof of Proposition 3.5.

Clearly (2) implies (3).

Assume that (3) holds. For $g\in G$ and $h^{\prime}\in H^{\prime}$, if
$\psi\circ g=h^{\prime}\circ\psi$, then $\psi\circ
g^{-1}={h^{\prime}}^{-1}\circ\psi$, and thus for any $g^{\prime}\in G(\psi)$,
we have

$\displaystyle\psi\circ(g^{-1}g^{\prime}g)$ $\displaystyle=(\psi\circ
g^{-1})\circ g^{\prime}g=({h^{\prime}}^{-1}\circ\psi)\circ g^{\prime}g$
$\displaystyle={h^{\prime}}^{-1}\circ(\psi\circ g^{\prime})\circ
g={h^{\prime}}^{-1}\circ\psi\circ g$
$\displaystyle={h^{\prime}}^{-1}\circ(\psi\circ
g)={h^{\prime}}^{-1}h^{\prime}\circ\psi=\psi.$

This means that $g^{-1}g^{\prime}g\in G(\psi)$. Thus (1) holds.

Assume that (1) holds. By the same argument of the last part of the proof of
Theorem 1.1, $\theta$ is a finite harmonic morphism of degree
$\frac{|G|}{|G(\psi)|}$. Here, $G(\psi)$ corresponds to $G_{1}$ and $G$ to
$G_{2}$ in this case. We define an isomertic action of $G/G(\psi)$ on
$\varGamma^{\prime\prime}$ as $[g](x^{\prime\prime}):=\psi(g(x))$ for $[g]\in
G/G(\psi)$, $x^{\prime\prime}\in\varGamma^{\prime\prime}$ and any
$x\in\psi^{-1}(x^{\prime\prime})$. It is well-defined. In fact, if
$x_{1}\in\psi^{-1}(x^{\prime\prime})$, then there exists $g_{1}\in G(\psi)$
such that $g_{1}(x_{1})=x$. Since $G(\psi)$ is a normal subgroup of $G$, there
exists $g_{2}\in G(\psi)$ such that $g(x)=gg_{1}(x_{1})=g_{2}g(x_{1})$. Thus
we have $\psi(g(x))=\psi(g_{2}g(x_{1}))=\psi(g(x_{1}))$. If
$[g]=[\widetilde{g}]$, then there exists $g_{1}\in G(\psi)$ such that
$g=g_{1}\widetilde{g}$. Hence
$\psi(g(x))=\psi(g_{1}\widetilde{g}(x))=\psi(\widetilde{g}(x))$. Since the
action of $G$ on $\varGamma$ is Galois, there exists a finite subset
$U^{\prime}$ of $\varGamma/G$ such that for any
$x^{\prime}\in(\varGamma/G)\setminus U^{\prime}$,
$|\pi^{-1}_{G}(x^{\prime})|=|G|$ holds. Let $O$ be the finite subset
$(\pi_{G/G(\psi)}\circ\psi)(\pi^{-1}_{G}(U^{\prime}))$ of
$\varGamma^{\prime\prime}/(G/G(\psi))$, where
$\pi_{G/G(\psi)}:\varGamma^{\prime\prime}\to\varGamma^{\prime\prime}/(G/G(\psi))$
is the natural surjection. Let
$[x^{\prime\prime}]\in(\varGamma^{\prime\prime}/(G/G(\psi)))\setminus O$. We
have $\pi^{-1}_{G/G(\psi)}([x^{\prime\prime}])=(G/G(\psi))x^{\prime\prime}$.
Since $|\varphi^{-1}(\theta(x^{\prime\prime}))|=|G|$ and $\psi$ is
$G(\psi)$-Galois, we have

$\displaystyle|(G/G(\psi))x^{\prime\prime}|=|\psi(\varphi^{-1}(\theta(x^{\prime\prime})))|=\frac{|G|}{|G(\psi)|},$

and thus the action of $G/G(\psi)$ on $\varGamma^{\prime\prime}$ is Galois.
Let

$\displaystyle\phi:\varGamma^{\prime\prime}/(G/G(\psi))\to\varGamma^{\prime};\qquad\pi_{G/G(\psi)}(x^{\prime\prime})\mapsto\theta(x^{\prime\prime}),$

where $x^{\prime\prime}\in\varGamma^{\prime\prime}$. With any fixed element
$x\in\psi^{-1}((G/G(\psi))x^{\prime\prime})$, since

$\displaystyle\theta((G/G(\psi))x^{\prime\prime})=\theta(\psi(Gx))=\theta(\psi(x))=\varphi(x),$

we have $\theta(x^{\prime\prime})=\varphi(x)$, and thus $\phi$ is well-
defined. By definition, for any $[g]\in G/G(\psi)$,
$\phi\circ\pi_{G/G(\psi)}=\theta\circ[g]$ holds. Since $\pi_{G/G(\psi)}$ and
$\theta$ are local isometries and $[g]$ is an automorphism, $\phi$ is a local
isometry. When $\theta(x^{\prime\prime})=\theta(y^{\prime\prime})$, with any
$x\in\psi^{-1}((G/G(\psi))x^{\prime\prime})$ and
$y\in\psi^{-1}((G/G(\psi))y^{\prime\prime})$,
$(G/G(\psi))x^{\prime\prime}=\psi(Gx)=\psi(Gy)=(G/G(\psi))y^{\prime\prime}$
hold, and thus
$\pi_{G/G(\psi)}(x^{\prime\prime})=\pi_{G/G(\psi)}(y^{\prime\prime})$ holds.
This means that $\phi$ is injective and hence a finite harmonic morphism of
degree one. In conclusion, $\theta$ is $G/G(\psi)$-Galois.

We shall show the last assertion. By (1), the quotient group $G/G(\psi)$
naturally acts on $\varGamma^{\prime\prime}$ as above. Since each stabilizer
subgroup of $G$ with respect to each point of $\varGamma$ except a finite
number of points is trivial, the natural action is faithful, i.e., for any
equivalence class $[g]\in G/G(\psi)$ other than $[1]$, there exists a point
$x^{\prime\prime}\in\varGamma^{\prime\prime}$ such that
$[g](x^{\prime\prime})\not=x^{\prime\prime}$. For any $g\in G$, $\psi\circ
g=[g]\circ\psi$ and there exists $h\in H$ such that $\psi\circ g=h\circ\psi$.
Since the natural action above is faithful, if $[g]\not=[\widetilde{g}]$, then
$[g]\circ\psi\not=[\widetilde{g}]\circ\psi$. Thus, the map

$\displaystyle G/G(\psi)\to H;\qquad[g]\mapsto h$

is injective. Since

$\displaystyle|H|=\operatorname{deg}(\theta)=\frac{|G|}{|G(\psi)|}=|G/G(\psi)|$

holds, this map is surjective. By definition, it is a group homomorphism. In
conclusion, we have the desired group isomorphism. ∎

To remove the last conditions in (2) and (3) of Proposition 3.5, we peove the
following lemma by using Proposition 3.4:

######  Lemma 3.6.

In the same setting in Proposition 3.5, the following are equivalent:

$(1)$ there exists a finite group $H$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H$-normal, and

$(2)$ there exists a finite group $H^{\prime}$ isometrically acting on
$\varGamma^{\prime\prime}$ such that $\theta$ is $H^{\prime}$-normal and for
any $g\in G$, there exists $h^{\prime}\in H^{\prime}$ satisfying $\psi\circ
g=h^{\prime}\circ\psi$.

###### Proof.

(2) clearly implies (1).

Assume that (1) holds. For any $g\in G$, we define
$h^{\prime}_{g}:\varGamma^{\prime\prime}\to\varGamma^{\prime\prime};\psi(x)\mapsto(\psi\circ
g)(x)$, where $x\in\varGamma$. Then, by Proposition 3.4, $h^{\prime}_{g}$ is a
local isometry. Since $\psi\circ g$ is surjective and continuous,
$h^{\prime}_{g}$ is an automorphism of $\varGamma^{\prime\prime}$. Set
$H^{\prime}$ as the group generated by these $h^{\prime}_{g}$. Then
$H^{\prime}$ induces an isometric and faithful action on $\varGamma^{\prime}$.
For any $x\in\varGamma$, we check that $H^{\prime}\psi(x)=H\psi(x)$. Since for
any $g\in G$, there exists $h\in H$ such that

$\displaystyle(h^{\prime}_{g}\circ\psi)(x)=(\psi\circ g)(x)=(h\circ\psi)(x),$

we have $H^{\prime}\psi(x)\subset H\psi(x)$. Conversely, for any $h\in H$,
since

$\displaystyle\varphi(\psi^{-1}((h\circ\psi)(x)))=\theta((h\circ\psi)(x))=(\theta\circ\psi)(x)=\varphi(x),$

$\psi^{-1}((h\circ\psi)(x))$ is contained in $Gx$. Therefore, for any
$\widetilde{x}\in\psi^{-1}((h\circ\psi)(x))$, there exists $g\in G$ such that
$\widetilde{x}=g(x)$. Thus, we have

$\displaystyle(h\circ\psi)(x)=\psi(\widetilde{x})=\psi(g(x))=(\psi\circ
g)(x)=(h^{\prime}_{g}\circ\psi)(x),$

and hence $H^{\prime}\psi(x)\supset H\psi(x)$. From this, the action of
$H^{\prime}$ on $\varGamma^{\prime}$ induces a transitive action on each fiber
by the transitivity of the action of $H$ on each fiber. In conclusion,
$\theta$ is $H^{\prime}$-normal and satisfies that $\theta\circ\psi=\varphi$
and for any $g\in G$, there exists $h^{\prime}_{g}\in H^{\prime}$ such that
$\psi\circ g=h^{\prime}_{g}\circ\psi$. ∎

By Proposition 3.5 and Lemma 3.6, we prove theorem 1.2.

Finally, we shall prove Theorem 1.3. This theorem means that all Galois
coverings are categorical quotients.

###### Proof of Theorem 1.3.

Since $\varGamma^{\prime}$ has the quotient topology, we have a unique
continuous map $\theta$ satisfying $\psi=\theta\circ\varphi$, and thus it is
enough to check that $\theta$ is a finite harmonic morphism. Fix loopless
models $(V,E,l),(V^{\prime},E^{\prime},l^{\prime})$ and
$(V^{\prime\prime},E^{\prime\prime},l^{\prime\prime})$ for
$\varGamma,\varGamma^{\prime}$ and $\varGamma^{\prime\prime}$, respectively,
compatible with $\varphi$ and $\psi$. We can choose such loopless models by
the assumptions. Let $e\in E$. As for any $x,y\in e$,
$\operatorname{dist}(x,y)=\operatorname{dist}(\varphi(x),\varphi(y))$ holds,
we have
$\operatorname{deg}_{e}(\psi)=\operatorname{deg}_{\varphi(e)}(\theta)$.
Therefore, $\theta$ is a finite morphism. Let $v$ be in $V$ and fix an edge
$e^{\prime\prime}\in E^{\prime\prime}$ incident to
$\psi(v)=\theta(\varphi(v))$. Then, since $\psi$ is a finite harmonic
morphism, we have

$\displaystyle\operatorname{deg}_{v}(\psi)$ $\displaystyle=\sum_{e\in
E:\,e\mapsto e^{\prime\prime},\,v\in e}\operatorname{deg}_{e}(\psi)$
$\displaystyle=\frac{\sum_{e\in E:\,\varphi(e)\mapsto
e^{\prime\prime},\,\varphi(v)\in\varphi(e)}\operatorname{deg}_{\varphi(e)}(\theta)}{|Gv|}$
$\displaystyle=\frac{\sum_{e^{\prime}\in E^{\prime}:\,e^{\prime}\mapsto
e^{\prime\prime},\,\varphi(v)\in
e^{\prime}}\operatorname{deg}_{e^{\prime}}(\theta)\cdot|\varphi^{-1}(e^{\prime})|}{|Gv|}$
$\displaystyle=\frac{|G|}{|Gv|}\cdot\left(\sum_{e^{\prime}\in
E^{\prime}:\,e^{\prime}\mapsto e^{\prime\prime},\,\varphi(v)\in
e^{\prime}}\operatorname{deg}_{e^{\prime}}(\theta)\right).$

Thus we have

$\displaystyle\sum_{e^{\prime}\in E^{\prime}:\,e^{\prime}\mapsto
e^{\prime\prime},\,\varphi(v)\in
e^{\prime}}\operatorname{deg}_{e^{\prime}}(\theta)=\operatorname{deg}_{v}(\psi)\cdot\frac{|Gv|}{|G|}=\frac{\operatorname{deg}_{v}(\psi)}{|G_{v}|}.$

It is independent of the choice of $e^{\prime\prime}$ and since
$\operatorname{deg}_{e^{\prime}}(\theta)=\operatorname{deg}_{e}(\psi)$ with
any $e\in E$ such that $\varphi(e)=e^{\prime}$, it is a positive integer.
These mean that $\theta$ is harmonic as $v$ is an element of $V$. The
uniqueness comes from the universal mapping property of quotient topology. ∎

######  Remark 3.7.

In [2], the author gave another definition of Galois coverings on tropical
curves. With that definition, for a map between tropical curves
$\varphi:\varGamma\to\varGamma^{\prime}$, if there exists an isometric action
of a finite group $G$ on $\varGamma$ such that there exists a finite harmonic
morphism of degree one $\theta:\varGamma/G\to\varGamma^{\prime}$ satisfying
$\varphi\circ g=\theta\circ\pi_{G}$ for any $g\in G$, we call $\varphi$ a
$G$-Galois covering on $\varGamma^{\prime}$. In this paper, let us call it a
$G$-preGalois covering on $\varGamma^{\prime}$ or a preGalois covering simply.
There exists a preGalois covering on a tropical curve for which the Galois
correspondence and the universal mapping property do not hold. See Examples
3.8, 3.9, and 3.10. By Examples 3.9 and 3.10, we know that for a preGalois
covering $\varphi:\varGamma\to\varGamma^{\prime}$, even there exists a model
$(G,l)$ for $\varGamma$ such that the greatest common divisor of
$\\{\operatorname{deg}_{e}(\varphi)\,|\,e\in E(G)\\}$ is one, $\varphi$ may
not be a Galois covering.

Figure 1. On each figure, black dots (resp. lines) stand for vertices (resp.
edges).

######  Example 3.8.

Let $G$ be the graph whose set of vertices consists of two vertices and whose
set of edges consists of three multiple edges $\\{e_{1},e_{2},e_{3}\\}$
between them (the left figure of Figure 1). Let $l$ be the length function
such that $l(E(G))=\\{1\\}$ and $\varGamma$ the tropical curve obtained from
$(G,l)$. Then, the symmetric group of degree three $\Sigma_{3}$ isometrically
acts on $\varGamma$ in a natural way. The quotient graph $G/\Sigma_{3}$ and
the length function
$E(G/\Sigma_{3})\to\boldsymbol{R}\cup\\{\infty\\};[e_{i}]\mapsto 2$ give the
quotient space $\varGamma/\Sigma_{3}$ a tropical curve structure. Let
$\varGamma^{\prime}$ stand for this quotient tropical curve. Then, the natural
surjection $\pi_{\Sigma_{3}}:\varGamma\to\varGamma^{\prime}$ is a
$\Sigma_{3}$-preGalois covering on $\varGamma^{\prime}$. Note that
$\varGamma^{\prime}$ is isometric to the closed interval $[0,2]$.

Let $\sigma$ be the cyclic permutation of multiple edges $(e_{1}e_{2}e_{3})$.
For a finite harmonic morphism $\psi:\varGamma\to\varGamma^{\prime\prime}$, if
$\psi\circ\sigma=\psi$, then for any $\beta\in\Sigma_{3}$,
$\psi\circ\beta=\psi$ holds. Hence, Theorem 1.1 does not hold for
$\pi_{\Sigma_{3}}$.

$\pi_{\Sigma_{3}}$ is not $\Sigma_{3}$-Galois and does not have a universal
mapping property. In fact, since
$\operatorname{deg}_{e_{i}}(\pi_{\Sigma_{3}})=2$, $\pi_{\Sigma_{3}}$ is not
$\Sigma_{3}$-Galois. Since for any $\beta\in\Sigma_{3}$,
$\pi_{\langle\sigma\rangle}\circ\beta=\pi_{\langle\sigma\rangle}$ holds, there
exists a unique continuous map $\theta$ satisfying
$\pi_{\langle\sigma\rangle}=\theta\circ\pi_{\Sigma_{3}}$. Here,
$\langle\sigma\rangle$ stands for the subgroup of $G$ generated by $\sigma$.
On the other hand, since $\operatorname{deg}(\pi_{\Sigma_{3}})=6$ and
$\operatorname{deg}(\pi_{\langle\sigma\rangle})=3$, $\theta$ is not a finite
harmonic morphism. Hence, $\pi_{\Sigma_{3}}$ does not have a universal mapping
property.

######  Example 3.9.

Let $e_{1},\ldots,e_{6}$ be the edges of the star of six edges $S_{6}$, i.e.,
the complete bipartite graph $K_{1,6}$ (the center figure of Figure 1). Let
$l$ be the length function such that $l(E(S_{6}))=\\{1\\}$ and $\varGamma$ be
the tropical curve obtained from $(S_{6},l)$. Let $\sigma$ (resp. $\beta$) be
the cyclic permutation $(e_{1}e_{2})$ (resp. $(e_{3}e_{4}e_{5}e_{6})$) and $G$
the finite group generated by $\sigma$ and $\beta$. Then, the quotient graph
$S_{6}/G$ and the length function
$E(S_{6}/G)\to\boldsymbol{R}\cup\\{\infty\\};[e_{1}]\mapsto 4;[e_{3}]\mapsto
2$ give the quotient space $\varGamma/G$ a tropical curve structure. Let
$\varGamma^{\prime}$ stand for this quotient tropical curve. Then, the natural
surjection $\pi_{G}:\varGamma\to\varGamma^{\prime}$ is a $G$-preGalois
covering.

For a finite harmonic morphism $\psi:\varGamma\to\varGamma^{\prime\prime}$, if
$\psi\circ(\sigma\beta^{2})=\psi$, then $\psi\circ\beta^{2}=\psi$ holds.
Hence, Theorem 1.1 does not hold for $\pi_{G}$.

Let $H$ be the subgroup of $G$ generated by $\beta$. Then, the natural
surjection $\pi_{H}:\varGamma\to\varGamma/H$ is $H$-preGalois that is not
$H$-Galois and does not have a universal mapping property. In fact, since
$\operatorname{deg}_{e_{1}}(\pi_{H})=4$, $\pi_{H}$ is not $H$-Galois. Note
that the quotient graph $S_{6}/H$ is the tree consisting of three vertices and
two edges $[e_{1}],[e_{3}]$, and $S_{6}/H$ and the length function
$E(S_{6}/H)\to\boldsymbol{R}\cup\\{\infty\\};[e_{1}]\mapsto 4;[e_{3}]\mapsto
1$ give the quotient space $\varGamma/H$ a tropical curve structure. Let
$\gamma$ be the cyclic permutation $(e_{1}e_{2}e_{3}e_{4}e_{5}e_{6})$. Then,
the quotient tropical curve $\varGamma/\langle\gamma\rangle$ is isometric to
the closed interval $[0,1]$. Since
$\pi_{\langle\gamma\rangle}\circ\beta=\pi_{\langle\gamma\rangle}$ holds, there
exists a unique continuous map $\theta$ satisfying
$\pi_{\langle\gamma\rangle}=\theta\circ\pi_{H}$. On the other hand, since
$\operatorname{deg}(\pi_{H})=4$ and
$\operatorname{deg}(\pi_{\langle\gamma\rangle})=6$, $\theta$ is not a finite
harmonic morphism. Hence, $\pi_{H}$ does not have a universal mapping
property.

######  Example 3.10.

Let $e_{1},\ldots,e_{5}$ be the edges of the star of five edges $S_{5}$, i.e.,
the complete bipartite graph $K_{1,5}$ (the right figure of Figure 1). Let $l$
be the length function such that $l(E(S_{5}))=\\{1\\}$ and $\varGamma$ be the
tropical curve obtained from $(S_{5},l)$. Let $\sigma$ (resp. $\beta$) be the
cyclic permutation $(e_{1}e_{2})$ (resp. $(e_{3}e_{4}e_{5})$) and $G$ the
finite group generated by $\sigma$ and $\beta$. Then, the quotient graph
$S_{5}/G$ and the length function
$E(S_{5}/G)\to\boldsymbol{R}\cup\\{\infty\\};[e_{1}]\mapsto 3;[e_{3}]\mapsto
2$ give the quotient space $\varGamma/G$ a tropical curve structure. Let
$\varGamma^{\prime}$ stand for this quotient tropical curve. Then, the natural
surjection $\pi_{G}:\varGamma\to\varGamma^{\prime}$ is a $G$-preGalois
covering.

$\pi_{G}$ is not $G$-Galois and does not have a universal mapping property. In
fact, since $\operatorname{deg}_{e_{i}}(\pi_{G})\geq 2$, $\pi_{G}$ is not
$G$-Galois. Let $\varGamma^{\prime\prime}$ be the tropical curve isometric to
the closed interval $[0,1]$. Note that $\varGamma^{\prime\prime}$ is the
quotient tropical curve of $\varGamma$ by the natural action of the group
generated by the cyclic permutation $\gamma:=(e_{1}e_{2}e_{3}e_{4}e_{5})$ on
$\varGamma$. Let
$\pi_{\langle\gamma\rangle}:\varGamma\to\varGamma^{\prime\prime}$ be the
natural surjection. Since for any $\delta\in G$,
$\pi_{\langle\gamma\rangle}\circ\delta=\pi_{\langle\gamma\rangle}$, there
exists a unique continuous map $\theta$ satisfying
$\pi_{\langle\gamma\rangle}=\theta\circ\pi_{G}$. On the other hand, since
$\operatorname{deg}(\pi_{\langle\gamma\rangle})=6$ and
$\operatorname{deg}(\pi_{\langle\gamma\rangle})=5$, $\theta$ is not a finite
harmonic morphism. Hence, $\pi_{G}$ does not have a universal mapping
property.

Note that since Galois coverings are preGalois coverings, for our definition
of Galois coverings, all corresponding assertions in [2] hold and in addition,
we need not consider tropical curves with edge-multiplicities.

## References

  * [1] Song JuAe, Generators of invariant linear system on tropical curves for finite isometry group, Hokkaido Mathematical journal 50(1):55–76, 2021.
  * [2] Song JuAe, Galois quotients of metric graphs and invariant linear systems, arXiv:1901.09172.
  * [3] Song JuAe, Galois actions for semifield extensions and Galois coverings on tropical curves, in preparation.
  * [4] Clive Reis and Stuart A Rankin, Abstract Algebra: Introduction to Groups, Rings and Fields with Applications (Second Edition), WSPC, 2013.