Understanding some square root notation in LMFDB occuring in the definition of a p-adic field

Question

I looked at the LMFDB page for the following $p$-adic field:

There I noted the notation $\mathbb{Q}_3(\sqrt{*})$ under "unramified subfield". Could you explain me what $*$ is supposed to be?

I think the square root makes sense as the extension is of degree $2$, so it is cyclic. Maybe the square root should just signify that the extension is cyclic but I am not sure.

Answer 1

That notation only seems to be used in the LMFDB for quadratic extensions. I don't see anywhere in the LMFDB where the meaning of this notation is spelled out; I'll ask around and see if we can add a line somewhere explaining it. Anyway, to work out what it means, let's look at the structure of quadratic extensions of $\newcommand{\QQ}{\mathbb{Q}}\newcommand{\ZZ}{\mathbb{Z}}\QQ_p$.

As with any field not of characteristic $2$, quadratic extensions of $\mathbb{Q}_p$ correspond to non-identity elements of $\QQ_p^\times/(\QQ_p^\times)^2$. So, let's look at the structure of this group. For $p \neq 2$, we have $$\QQ_p^\times \cong \underbrace{p^\ZZ \times \mu_{p-1} \times (1 + p\ZZ_p)}_{\text{written multiplicatively}} \cong \underbrace{\ZZ \times (\ZZ/(p-1)\ZZ) \times \ZZ_p}_{\text{written additively}},$$ where the isomorphism is given by writing each $x \in \QQ_p$ in the form $p^n \zeta (1 + py)$, where $\zeta^{p-1} = 1$. For $p = 2$, we have the slight variant $$\QQ_2^\times \cong \underbrace{2^\ZZ \times \{\pm 1\} \times (1 + 4 \ZZ_2)}_{\text{written multiplicatively}} \cong \underbrace{\ZZ \times (\ZZ/2\ZZ) \times \ZZ_2}_{\text{written additively}}.$$ So we have $$\QQ_p^\times/(\QQ_p^\times)^2 \cong \begin{cases} (\ZZ/2\ZZ)^2 & \text{if } p \neq 2, \\ (\ZZ/2\ZZ)^3 & \text{if } p = 2. \end{cases}$$

In particular, for $p \neq 2$, there are three quadratic extensions of $\QQ_p$: $$\QQ_p(\sqrt{*}), \quad \QQ_p(\sqrt{p}), \quad \QQ_p(\sqrt{p*}),$$ where $*$ stands in for any non-square in $\ZZ_p^\times$ (they all produce the same quadratic extensions). The extension $\QQ_p(\sqrt{*})$ is unramified and the other two are ramified.

The case for $p = 2$ is similar except that we have seven extensions: $$\QQ_2(\sqrt{*}), \QQ_2(\sqrt{2}), \QQ_2(\sqrt{-1}), \QQ_2(\sqrt{-2}), \QQ_2(\sqrt{2*}), \QQ_2(\sqrt{-*}), \QQ_2(\sqrt{-2*}),$$ where $*$ stands in for any non-square in $1 + 4\ZZ_2$. The first of these is unramified and the rest are ramified.

Answer 2

The unramified subfield of $K/\Bbb{Q}_p$ is $\Bbb{Q}_p(\zeta_{p^f-1})$ where $p^f$ is the cardinality of $O_K/m_K$. It is also $\Bbb{Q}_p(a)$ for any $a$ with a monic $\Bbb{Z}_p[x]$ minimal polynomial $h$ such that $\Bbb{F}_{p^f} =\Bbb{F}_p[x]/(h)$.

At the bottom of your link they gave an example of $h \in \Bbb{Z}[x]$, because the minimal polynomial of $\zeta_{p^f-1}$ doesn't have coefficients in $\Bbb{Q}$.