In Field Theory, some basic problems

Question

$\def\Fp{\mathbb F_p}$ 1. Determine whether the following statements are True of False. give brief reasons.

(A) Let $u$ and $v$ are indeterminates. The field $\Fp(u,v)$ has a primitive element over $\Fp(u^p, v^p)$.

(B) for any $n\ge 1$, there exists a field extension of $\Fp$ of degree $n$.

(C) Let $\mathrm{char}\;F = p$ and $f \in F[x]$. Then $f$ is inseparable (or not separable) over $F$ if and only if $f \in F[x]^p$.

(D) The polynomial $x^4 + 1$ is irreducible over $\mathbb F_3$.

(E) Let $F < K < L$ be field extensions. Then $[K:F]$ divides the other of the group $\mathrm{Gal}(L|F)$.

(F) Let $a$,$b$ be the two distinct roots of $x^3 + x^2 + 2x + 1$ over $\mathbb Q$. Then $\mathbb Q < \mathbb Q (a, b)$ is Galois.

Answer 1

For problem (A) I suggest you look at theorems concerning infinitely many intermediate extensions (I think you will find this in Dummit and Foote concerning composite and simple extensions), problems (B) is trivial, problem (D) has been discussed extensively on this site ($x^4 +1$ is reducible over $\Bbb{F}_p$ for all primes $p$) while (E) does not make sense (what if $L/K$ is not a Galois extension?).