What finite groups G can appear as collections of automorphisms of some field? More precisely, for which G does there exist a field F such that G is a subgroup of the automorphism group of F?
What finite groups G can occur as collections of automorphisms of some finite field F?
Here are my proof ideas
- Recall that every finite group G is a subgroup of $S_n$, where n is the order of G. We claim that every finite group can appear as a collection of automorphisms of some field.
Let G be a group of order n. Then there exists a polynomial p(x) in $\mathbb{Q}[x]$ such that the splitting field for p(x), L, has Galois group $S_n$. Then corresponding to every subgroup H of $S_n$ is a fixed field $L^H$ and H is the Galois group of $L^H$. Therefore G is the Galois group of its fixed field $L^G$. Note that the Galois group is a group of automorphisms and therefore G is the collection of automorphisms of $L^G$.
- We claim that every finite cyclic group can occur as collections of automorphisms of some finite field.
Consider the cyclic group $G=\mathbb{Z}/n\mathbb{Z}$ and the finite field F of order p. Then the finite field L of order $p^n$ is the splitting field over F of the polynomial $x^{p^n}-x$. Then L over F is a Galois extension with Galois group G. Moreover, we know that every Galois group of a finite extension of a finite field is cyclic.
Questions 1. Are my proof ideas correct? 2. How do we actually find a polynomial p(x) with Galois group $S_n$ for every n? 3. For problem 2, what if the extension is infinite?
