For a finite Galois extension $M / K$ we denote by $Γ(M: K)$ the Galois Group of $M$ over $K$.

Let $K$ be a field. Let $M$ be a field containing $K$. Let $L_1, L_2$ be subfields of $M$.

For any field $F$, let us denote by $F(t)$ the field of rational functions in the variable $t$ over the field $F$. Let $𝔽_{125}$ be the finite field of order 125, let $K=𝔽_{125}(t)$, and let $f=x^{124}-t∈K[x]$. In this exercise, you can assume without proof that $f$ is irreducible in $K[x]$.

Let $n$ be a positive integer and let $G$ be a subgroup of the symmetric group $S_n$, acting on $\{1, …, n\}$ on the right. Let $ℚ$ be the field of rational numbers. Let $L=ℚ(x_1, … x_n)$ be the field of rational functions in the variables $x_1, … x_n$. Consider the action of $G$ on $L$ by relabelling the variables, i.e. for any $g ∈ G$ and any $f(x_1, … x_n) ∈ L, f^g(x_1,…,x_n)=f(x_{1 g}, …, x_{n g})$. Write $L^G=\{f ∈ L: f^g=f\text{ for all }g ∈ G\}$.


Let $G$ be a finite group acting faithfully on $K$. Then the extension $K/K^G$ is Galois, we have $[K:K^G]=|G|$, and the Galois group of the extension is $G$.

Lemma 9.21.6.

Pick finitely many elements $α _ i ∈ K$, $i = 1, … , n$ such that $σ (α _ i) = α _ i$ for $i = 1, … , n$ implies $σ $ is the neutral element of $G$. Set \[ L = K^ G(\{ σ (α _ i); 1 ≤ i ≤ n, σ ∈ G\} ) ⊂ K \] and observe that the action of $G$ on $K$ induces an action of $G$ on $L$. We will show that $L$ has degree $|G|$ over $K^ G$. This will finish the proof, since if $L ⊂ K$ is proper, then we can add an element $α ∈ K$, $α ∉ L$ to our list of elements $α _1, … , α _ n$ without increasing $L$ which is absurd. This reduces us to the case that $K/K^ G$ is finite which is treated in the next paragraph.

Assume $K/K^G$ is finite. By Lemma 9.19.1 we can find $α ∈ K$ such that $K = K^G(α)$. By the construction in the first paragraph of this proof we see that $α $ has degree at most $|G|$ over $K$. However, the degree cannot be less than $|G|$ as $G$ acts faithfully on $K^G(α) = L$ by construction and the inequality of Lemma 9.15.9.