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Edit : My question has been linked with this:Quadratic extension of field of characteristic 2 can/cannot be obtained by adjoining a single square root

and this: Field Extensions of degree $2$ but none of this answers (a).

This question is from my abstract algebra assignment and I was unable to solve it and so looking for help here.

(a) Let F be a field of odd characterstic and let K be a field extension of F of degree 2. Prove that there exists an $a\in F$ such that $K\approx F(\sqrt a)$.

(b) Give an example of a degree 2 extension K of a field F of characterstic 2 which is not obtained by attaching a square root of an element of F.

Attempt :For (a) I took minimal polynomial equal to $ax^2+ px +q$ where a, p, q$\in F$ but I am unable to reason why the root of the polynomial must be of the form $\sqrt a$. Something related to odd characterstic should be used but I am unable to find.

(b) for (b) I tried by taking the field $\mathbb{Q}$ , $\mathbb{R}$ but was unable to construct an example.

Please give some hints?

So, please help.

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    This has been asked on mse before. See here and here for instance. As an aside, for part (b) you were asked to consider fields of characteristic $2$, whereas $\mathbb{Q}$ and $\mathbb{R}$ are fields of characteristic $0$. You'll want to consider extensions of $\mathbb{Z}/2$ (say). – HallaSurvivor Sep 05 '21 at 06:47
  • @HallaSurvivor None of these linked questions answer part(a) of my question. I request you to voto to reopen this question. –  Oct 17 '21 at 05:35
  • @HallaSurvivor Can you please elaborate on what should I do in case of $\mathbb{Z}/ 2$ extension fields? –  Oct 17 '21 at 05:49

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