1) Let $K/k$ an extension field. If $[K:k]=2$ show that there is $a\in K$ s.t. $K=k(\alpha )$ and $\alpha ^2=a$.
2) Does it work for $[K:k]=p$ when $p$ is prime ? Or for other number ? (like non-prime)
My work
For 1), if $[K:k]=2$ there is an $a\in K$ that is not in $k$. Let $X^2-\alpha $ its minimal polynomial. Then $$a^2-\alpha =0\implies (a-\sqrt \alpha )(a+\sqrt \alpha )=0\implies a=\pm\sqrt \alpha $$
It looks to be very confusing... any idea ?
You have to be careful in the logical order of your arguments here.
It is correct to say that $[K:k] = 2$ implies there is an $a \in K \setminus k$. If its minimal polynomial is $X^2 - \alpha$ for $\alpha \in k$ then you certainly have that $a^2 - \alpha = 0$. This implies that $a^2 = \alpha$. This means that $a$ is a square root of $\alpha$ and that $-a$ is another (indeed, the only other) square root of $\alpha$ in $K$. At this point, however, the symbol $\sqrt{\alpha}$ is meaningless, believe it or not. You have not chosen any embedding of $K$ into a field where $\sqrt{\alpha}$ has already been defined (for example, some algebraic closure of $k$) and so it is not yet actually defined. You could define $\sqrt{\alpha} := a$ in $K$ and then you would of course have $-\sqrt{\alpha} = -a$ but there is little to be gained from this.
Instead of all of that, since it is clear that there is some element in $K\setminus k$ (namely $a$), your only mission is to show that in fact $a$ generates the extension - i.e. $K = k(a)$. This will be true because $K$, being a finite field extension of $k$ of degree 2, is necessarily a $k$-vector space of dimension $2$. Since $a \not\in k$, it is linearly independent from $1 \in k$ and so $\{1,a\}$ is a maximally linearly independent set in the vector space $K$ and is thus a basis. This means every element $b \in K$ can be written as $b = c_1\cdot 1 + c_2\cdot a$ for $c_1, c_2 \in k$. Hence $K = k(a)$.
As for your second question, I think the answer to be given here is that for 'nice' extensions (those being finite, separable ones), the statement remains true. That is, an extension $K/k$ is generated by a single element (called a 'primitive element'). This result is known as the Primitive Element Theorem. The caution is that for extension degree higher than 2, a primitive element need not be a $d^{th}$ root of an element of the base field (for $d$ the degree of the extension).