The G-Fixed Subfield of a Field

The G-Fixed Subfield of a Field

Theorem 1: Let $F$ be a field and let $G$ be a subgroup of $\mathrm{Aut}(F)$. The set $F^G = \{ a \in F : \theta(a) = a \: \mathrm{for \: all \:} \theta \in G \}$ is a subfield of $F$.
  • Proof: Let $a, b \in F^G$ and let $\theta \in G$. Then:
(1)
\begin{align} \quad \theta(a + b) = \theta(a) \pm \theta(b) = a + b \end{align}
(2)
\begin{align} \quad \theta(ab) = \theta(a) \theta(b) = ab \end{align}
  • So $a + b \in F^G$ and $ab \in F^G$. Also:
(3)
\begin{align} \quad \theta(-a) = -\theta(a) = -a \end{align}
(4)
\begin{align} \quad \theta(a^{-1}) = [\theta(a)]^{-1} = a^{-1} \end{align}
  • Therefore $-a \in F^G$ and $a^{-1} \in F^G$. So $F^G$ is a subfield of $F$. $\blacksquare$

The field above is given a special name which we define below.

Definition: Let $F$ be a field and let $G$ be a subgroup of $\mathrm{Aut}(F)$. The $G$-Fixed Subfield of $F$ is the subfield $F^G = \{ a \in F : \theta(a) = a \: \mathrm{for \: all \:} \theta \in G \}$.

For example, consider the field $\mathbb{C}$ of complex numbers. Let $\theta : \mathbb{C} \to \mathbb{C}$ be the automorphism defined for all $a + bi \in \mathbb{C}$ by:

(5)
\begin{align} \quad \theta(a + bi) = a - bi \end{align}

Let $G = \{ \mathrm{id}_{\mathbb{C}}, \theta \}$. Then it is easy to see that $\mathbb{R}$ is the $G$-fixed subfield of $F$.

Theorem 2: Let $K$ be a field and let $F$ be the splitting field of a separable polynomial over $K$. Let $G = \mathrm{Gal} (F/K)$. Then $F^G = K$.
  • Proof: We have that:
(6)
\begin{align} \quad K \subseteq F^G \subseteq F \end{align}
  • Since $F$ is a splitting field of a separable polynomial over $K$. Also $F$ is a splitting field of that separable polynomial over $F^G$. Therefore:
(7)
\begin{align} \quad [F : K] = |\mathrm{Gal}(F/K)| = |\mathrm{Gal}(F/E)| = [F : F^G] \end{align}
  • Now observe that
(8)
\begin{align} \quad [F : K] = [F : F^G][F^G : K] \end{align}
  • Hence $[F^G : K] = 1$. So $F^G = K$. $\blacksquare$
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