(A) Let $F$ be a finite field extension of $\mathbb{Q}$. Show that any field homomorphism $\phi : F \rightarrow F$ is an isomorphism. (Note that $\phi(1) = 1$ by definition.)\\
(B) Let $F$ be a finite field whose characteristic is not 2. Let $F^\times$ denote the multiplicative group of nonzero elements of $F$. An element $a \in F^\times$ is called a square if there exists $x \in F^\times$ such that $x^2 = a$. Show that exactly half the elements of $F^\times$ are squares.