Automorphism Group of a Field Extension

Subjects: Field Theory
Links: Field Extensions, Automorphism Group, Classification of Simple Field Extensions

An isomorphism between two extension $L/F$ and $K/F$ over the same field, is an field isomorphism $\varphi :L\to K$ such that $\varphi {|}_F={\text{id}}_F$, i.e., the following diagram commutes

\usepackage{tikz-cd} 
\usepackage{amsfonts, amsmath, amssymb}
\begin{document} 
\begin{tikzcd}[row sep=2cm, column sep=2cm]    
L\arrow[r, "\phi"]\arrow[d, dash] & K\arrow[d, dash]\\
F\arrow[r, "\text{id}_F"] & F
\end{tikzcd}
\end{document}

In particular, it means that $\varphi$ fixes $F$. Thus, an automorphism of field extension $K/F$, is a field automorphism $\varphi :K\to K$ such that $\varphi {|}_F={\text{id}}_F$, i.e., a field automorphism of $K$ that leave $F$ fixed. The automorphisms of the field extension $K/F$ are called $F$-automorphisms of $K$.

Obs: If $K/F$ is a field extension, then the set of $F$-automorphisms of $K$, denoted $\text{Aut}(K/F)$ is a group, and it is a subgroup of $\text{Aut}(K)$.

Def: Let $L/K$ and $K/F$ be a field extensions. I am gonna use the notation $$S(K) := \text{Aut}(L/K) \le \text{Aut}(L/F),$$to denote the subgroup $\text{Aut}(L/F)$ associated with the intermediate field $K$. We have the function $$S: \mathcal F_{L/F}:={\text{Intermediate fields of }L/F}\to \mathcal G_{L/F}:={\text{Subgroups of }\text{Aut}(L/F)}.$$
We have the following properties of this function $S$.

Prop: The function $S:{\mathcal{C}}_{L/F}\to {\mathcal{G}}_{L/F}$ satisfies the following:

• If $M_1\subseteq M_2$ are intermediate fields, then $S$ flips the inclusions, i.e., $S(M_2)\subseteq S(M_1).$
• For the intermediate field, $M=F$, we have that $S(M)=\text{Aut}(L/F)$ is the total group.
• For the intermediate field $M=L$, we have that $S(L)=\text{Aut}(L/L)=\{{\text{id}}_L\}$ is the trivial subgroup.

Given a subgroup $H$ of the group $G=\text{Aut}(L/F)$ we would like to associate it with an intermediate field of $L/F$.

Let us observer that $\text{Aut}(L)$ acts naturally on $L$, since the action is just $(\sigma ,\alpha )\in \text{Aut}(L)\times L\mapsto \sigma (\alpha )\in L$. This means that if $L/F$ is a field extension, then $\text{Aut}(L/F)$ also acts naturally on the extension $L/F$, by the same action.

Given $H\le \text{Aut}(L/F)$ let $$L^H := {a\in L \mid a\in \forall \sigma\in H(\sigma(a) = a) },$$meaning, $L^H$ is the subset of $L$ that ate fixed by all the elements of $H\le \text{Aut}(L/F)$. This is just the $H$-fixed points of $L$.

Lemma: If $H\le \text{Aut}(L/F)$, then $L^H$ is an intermediate field of $L/F$.

Def: If $H\le \text{Aut}(L/F)$, then $L^H$ is called the fixed field of the subgroup $H\le \text{Aut}(L/F)$. We have the function $$ F: \mathcal G_{L/F}\to \mathcal C_{L/F}$$ which for every subgroup $H$ of $\text{Aut}(L/F)$ we associate its corresponding fixed field $F(H):=L^H$.

Lemma: The function $F:{\mathcal{G}}_{L/F}\to {\mathcal{C}}_{L/F}$ satisfies the following:

• If $H_1\le H_2$ be subgroups of $\text{Aut}(L/F)$, then $F$ flips inclusions, i.e., $F(H_1)\supseteq F(H_2)$.
• If $H\le \text{Aut}(L/F)$, then $H\le S(F(H))$.
• If $M$ is an intermediate field of $L/F$, then $M\subseteq F(S(M))$.

Def: A character $\chi$ of a group $G$ with values in the field $L$ is a homomorphism from $G$ to the multiplicative group of $L$: $\chi :G\to L^{\times }$, i.e., $\chi (g_1{g}_2)=\chi (g_1)\chi (g_2)$ for all $g_1,g_2\in G$.

Def: The characters ${\chi }_1,\dots ,{\chi }_n$ of $G$ are said to be linearly independent over $L$ if they are linearly independent as function on $G$,i.e., there is no nontrivial relation $$a_1 \chi_1 + \dots + a_n \chi_n = 0$$with $a_1,\dots ,a_n\in L$ not all zero.

Linear Independence of Characters: If ${\chi }_1,\dots ,{\chi }_n$ are distinct characters of $G$ with values in $L$ then they are linearly independent over $L$ .

Dedekind theorem: Let $F$ be a field. Then, any finite set of distinct automorphisms ${\sigma }_1,\dots ,{\sigma }_n:F\to F$ are linearly independent. Meaning, if $a_1{\sigma }_1(x)+a_n{\sigma }_n(x)=0$ for all $x\in F$, then $a_i=0$ for all $i\in \{1,\dots ,n\}$.

Cor: If $L/F$ is a finite field extension, then $|\text{Aut}(L/F)|\le [L:F]$.

Th: Let $G=\{{\sigma }_1,\dots ,{\sigma }_n\}$ is a finite group of automorphisms of a field $L$ and $L^G$ is the fixed field of $G$, then $$[L: L^G]= |G| = n.$$
Artin Theorem: Let $G$ be a subgroup of a finite automorphism group of a field $L$. If $L^G$ is the fixed field of $G$, then $\text{Aut}(L/L^G)=G$.

Cor: Let $L$ be a field. The function $F$, that assigns each finite subgroup $G$ of $\text{Aut}(L)$ to $L^G,$ is injective.

The results above show that the importance for a finite extension $L/F$ is to have that $F=L^G$ is the fixed field of the group $\text{Aut}(L/F)$.