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Let $p$ be a prime number. If we want to judge whether $p$ is prime element of $ \Bbb{Q}_p(α)$ for some fixed element $α∈\overline{ \Bbb{Q}_p}$, what is the basic strategy?

To find a ring of integers of $ \Bbb{Q}_p(α)$ is much more difficult than to check $p$ is prime in $ \Bbb{Q}_p(α)$, to be precise, to check $p$ is prime of ring of integers of $ \Bbb{Q}_p(α)$

(c.f Ring of integers in p-adic field)

How can I check whether $p$ is prime in the ring of integers of $ \Bbb{Q}_p(α)$ without finding ring of integers ?

Pont
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  • Yes, thank you, that was my typo. – Pont May 01 '22 at 17:15
  • Algebraic integers could mean a traditional root of a monic polynomials in $\mathbb Z[x]$ or a generalized algebraic integer, extending the $p$-adic integers, a root of a monic polynomials in $\mathbb Z_p[x].$ – Thomas Andrews May 01 '22 at 17:17
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    The strategy is to construct the unramified extension $K$ of degree $[\Bbb{Q}_p(\alpha):\Bbb{Q}_p]$ and to use Hensel lemma to check if $\alpha$'s minimal polynomial has a root in $K$. – reuns May 01 '22 at 17:21
  • If you have the minimum polynomial of $\alpha$, isn't it enough to check that [$\Bbb{Q}_p[\alpha]:\Bbb{Q}_p]=[\Bbb{F}_p[\bar{\alpha}]:\Bbb{F}_p]$ – sharding4 May 01 '22 at 17:31
  • @reuns : $K$ is degree $[ \Bbb{Q}_p(α):\Bbb{Q}_p]$ over which field? $ \Bbb{Q}_p$? – Pont May 01 '22 at 17:46

1 Answers1

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Here is a useful property: if $f(x) \in \mathbf Z_p[x]$ is monic and separable mod $p$ and $\alpha$ is a root of $f(x)$, then $\mathbf Q_p(\alpha)$ is unramified over $\mathbf Q_p$ ($p$ is prime in the extension).

Example: if $p \nmid n$ then $x^n - 1 \bmod p$ is separable, then $\mathbf Q_p(\zeta)$ is unramified over $\mathbf Q_p$ for all $n$th roots of unity $\zeta$.

Note it is not important in the property that $f(x)$ be irreducible, which is why I was able to apply it to $f(x) = x^n - 1$ above; it is reducible for $n > 1$.

Other methods that can help determine ramification include Newton polygons or look here.

KCd
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