examples of unramified extensions of $\mathbb{Q}_p$

Question

For every local field $K$ and natural number $n$ coprime to $K$'s residue characteristic, there is a unique unramified extension $L/K$ of degree $n$.

Let's take $K=\mathbb{Q}_p$. What are some concrete examples of finite unramified extension of $\mathbb{Q}_p$? I suppose they are of the form $\mathbb{Q}_p[x]/(f(x))$ where $f(x)$ is irreducible over $\mathbb{Q}_p$, and perhaps mod $p$, $\overline{f(x)}$ is a factor of $x^q-x$? Perhaps I should run this backwards: start with an irreducible polynomial $g(x)$ over $\mathbb{F}_p$, pick a lift $G(x)$ to $g(x)$ to $\mathbb{Q}_p$. Will Hensel's lemma guarantee that $G(x)$ is irreducible over $\mathbb{Q}_p$? And then will $\mathbb{Q}_p[x]/(G(x))$ will be an unramified extension of $\mathbb{Q}_p$?

If you reference for where some examples are given, that is fine too.

Possible duplicate of unramified quadratic extensions of 2-adic numbers — Dietrich Burde, Jan 28 '16 at 16:23
@DietrichBurde Yes, I saw that while I was writing my question, and upvoted your answer, but the answer doesn't explain if any lift of g(x) is irreducible, etc. — usr0192, Jan 28 '16 at 16:30
As a quick example, $\mathbb{Q}_2(\sqrt{5})$ is the unique unramified extension of degree $2$. — Matt B, Jan 28 '16 at 17:35

score 12 · Accepted Answer · answered Jan 28 '16 at 22:39

You get unramified extensions of $\Bbb Q_p$ by adjoining roots of unity of order prime to $p$; alternatively, by adjoining $(p^n-1)$-th roots of unity.

The finite unramified extensions of $\Bbb Q_p$ are in natural one-to-one correspondence with the finite algebraic extensions of $\Bbb F_p$. This means that there’s only one unramified extension of $\Bbb Q_p$ of each degree. Want the cubic unramified extension of $\Bbb Q_3$? Find a cubic irreducible over $\Bbb F_3$, like $X^3-X-1$, and use the same polynomial’s roots over $\Bbb Q_3$.

examples of unramified extensions of $\mathbb{Q}_p$

1 Answers1

Linked