Applications of Algebraic Extensions (13.2.16-17)

Dummit and Foote Abstract Algebra, section 13.2, exercises 16-17:

16. Let $K/F$ be an algebraic extension and let $R$ be a ring where $F ⊆ R ⊆ K$. Show $R$ is a field.

17. Let $f(x)∈F[x]$ be irreducible of degree $n$, and let $g(x)∈F[x]$. Prove that every irreducible factor of $f(g(x))$ has degree divisible by $n$.

Proof: (16) It suffices to show that every element $r∈R$ has a multiplicative inverse. Since $r∈K$ is algebraic over $F$, we observe $r^{-1}∈F(r) ⊆ R$.

(17) Let $h(x) \mid f(g(x))$ be irreducible of degree $k$, and let $α$ be a solution to $h(x)$, so that $h(α)=0$ and thus $f(g(α))=0$. Since $f(x)$ is irreducible, we must have $g(α)$ is of degree $n$. We thus have $$\text{deg }h(x) = k = [F(α)~:~F] =$$ $$[F(α)~:~F(g(α))] \cdot [F(g(α))~:~F] = [F(α)~:~F(g(α))] \cdot n$$ so that $n \mid k$.$~\square$