# In Field Theory, some basic problems

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

(A) Let $u$ and $v$ be 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.

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In Computer voice: Computing ... does not compute ... give previously made attempts ... –  Thomas Jul 16 '12 at 12:35
This is not the way we do things here. First, don't order us to do things. Second, give us some idea of where you are with these problems. Do you know what the terms mean? Have you made any progress on them? What makes them interesting to you? Where did you come across them? You know, when you splash 6 unrelated questions up in one go, it makes people think you are trying to get someone to do your homework assignment for you - is that the impression you want to give? Ask one question; tell us what you know about it; digest the answers thoroughly; then ask another; etc. –  Gerry Myerson Jul 16 '12 at 12:48

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?).

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B is trivial? It's trivial that for all $p$ and all $n$ there's a polynomial of degree $n$ irreducible modulo $p$? It's not hard (and I'm sure it has been answered elsewhere on this site), but is it really trivial? –  Gerry Myerson Jul 17 '12 at 5:52
Also, I don't see why E doesn't make sense. Some sources use Gal(L/F) to mean the group of F-automorphisms of L even when L/F is not Galois. –  Gerry Myerson Jul 17 '12 at 5:56
@GerryMyerson I'm sorry, I misread that bit and thought it asked to show that a finite field always has $p^n$ elements. –  user38268 Jul 17 '12 at 12:26