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Show that:

a) The field Extension $\mathbb{F_p}(X,Y)|\mathbb{F_p}(X^p,Y^p)$ is not simple.

b)Find a primitive element of the field extension $\mathbb{Q}(\sqrt2+i,\sqrt3-i)|\mathbb{Q}$

c) Let $L|K$ be a separable field extension and there exists a $n\in \mathbb{N}$ so that $[K(x):K]\le n$ for all $x \in L$. Show that: $|L:K| \le n$.

Now in a) I don't have a clue. For be I could find the minimal polynoms, but I don't know how to continue... . It would be nice if you gave me some advice how to start. Thank you for taking your time.

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    You should perhaps only post one of the questions at a time. – 2017-01-10
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    For a) is $\mathbb{F}_p(X)/\mathbb{F}_p(X^p)$ a simple extension ? What is different with $\mathbb{F}_p(X,Y)/\mathbb{F}_p(X^p,Y^p)$ ? – 2017-01-10

2 Answers 2

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Hint for a): Show that the extension has infinitely many subfields, for $f\in K=\mathbb{F}_p(X,Y)$ consider the fields $K(fX+Y)$ and show that they are all non-isomorphic. Hence the extension cannot be simple.

Hint for b): Search MSE for this question, e.g., here.

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I wish to help you: for $2)$-$3)$

Theorem: Let $L$ field extension of $K$ and $u_1,u_2,...,u_n\in L$ that $u_1$ algebraic over $K$ , $u_2,...,u_n$ algebraic and separable over $K$ then exists $\gamma \in L$ that $K(\gamma)=F=K(u_1,u_2,...u_n)$ (i.e. $F$ simple extension of K ) and $\gamma$ is a primitive element.