Unramified extension of $L(\sqrt{\alpha})/L$ I am studying an article of Chaoli and I try to understand the following statement:
If $L$ is a number field and $\alpha \in L^{\times}/(L^{\times})^{2}$ then, for an odd prime $p$,  $L_{p}(\sqrt{\alpha})/L_{P}$ is unramified if and only if $\alpha$ has even valuation.
for $p=2$, $L_{2}(\sqrt{\alpha})/L_{2}$ is unramified if and only if $\alpha$ has even valuation and is represented by a $ unit \equiv 1\pmod{4}$
I try to use Kummer Theory but I don't know how can determine unramified  extension of a local field?!
 A: The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $\mu_p$ of $p$-th roots of unity, and let $L=K(\sqrt [p]\alpha)$ for some $\alpha \in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $v\nmid p $: The place $v$ is unramified in $L/K$ iff $v(\alpha) \equiv 0$ mod $p$ ; (ii) Wild case, $v\mid p$: Let $e_v$ be the ramification index of $v$ in $K/\mathbf Q(\mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v \in K^*$ s.t. $(\alpha/{x_v}^p)\equiv 1$ mod ${\frak P_v}^{pe_v}$.
Idea of proof : (i) is clear.  For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
