Galois extension of $\mathbb{Q}_p$ with Galois group $\mathbb{Z}/2\mathbb{Z}$

Question

Denote by $\mathbb{Q}_p$ the $p$-adic completion of the integers, i.e. with respect to the valuation $|x|=p^{-ord_p(x)}$. My question is, how do we find, and classify explicitly, all the field extensions $\mathbb{Q}_p\subset L$ with Galois group $G(L/\mathbb{Q}_p)=\mathbb{Z}/2\mathbb{Z}$?

Answer 1

By Kummer theory, such extensions bijects with nontrivial elements of $\mathbb Q_p^\times/(\mathbb Q_p^\times)^2$. Explicitly, $a\ne 1\in\mathbb Q_p^\times/(\mathbb Q_p^\times)^2$ corresponds to $\mathbb Q_p(\sqrt a)$.

The group can be described via the following exact sequence, for $p>2$: $$1\to \mathbb Z_p^\times/(\mathbb Z_p^\times)^2\cong\mathbb F_p^\times/(\mathbb F_p^\times)^2\to \mathbb Q_p^\times/(\mathbb Q_p^\times)^2\xrightarrow v\mathbb Z/2\to 0,$$ so $\mathbb Q_p^\times/(\mathbb Q_p^\times)^2\cong(\mathbb Z/2)^2$.