How to show the fixed field is a subfield

Question

Given and extension $K$ of a field $F$, and a subgroup $H \leq \text{Gal}(F/K)$, how can I show that the fixed field of $H$ given by: $$\{ x\in F : \varphi(x) = x\;\; \forall \varphi\in H \}$$ is a subfield of $F$ containing $K$?

Answer 1

You show that it contains 0, 1, and is closed under field operations ($+,-,\cdot,(\cdot)^{-1}$). These follow because we're looking at being fixed by a homomorphism.

Also showing that it contains $K$ follows by definition of the Galois group.