HiAll, I am stuck with this problem:
(a) Let $K$ be a field such that characteristic of $K$ is not 2. Prove that any extension $L$ of $K$ with $K\subset L$, and $[L:K]=2$ has the form $L=F(\beta)$ for some $\beta\in L^* \setminus K^*$ with $\beta^2\in K^*$.
(b) Is this true when the ground field $K$ has characteristic 2?
Here is my attempt for part (a): Choose an element $\beta\in L^*\setminus K^*$ such that $\{1,\beta\}$ is linearly independent over $K$.
Since $[L:K]=2$, so $\{1,\beta\}$ is a basis for $L$ over $K$.
Since $\beta$ cannot be written as $c_1 +c_2 \beta^2$, with $c_i\in K$ (not sure if this is true??), hence $\{1,\beta^2\}$ is not a basis and is hence linearly dependent.
Hence $\beta^2$ is a scalar multiple of $1$ and is hence in $K*$.
I am pretty sure it is not entirely right, as I didn't even make use of the fact that the characteristic of $K$ is not 2.
For (b), I think the answer is no, not true, but am not sure how to come up with an counterexample.
Sincere thanks for any help!