The **defining polynomial** of a $p$-adic field $K$ is an irreducible polynomial $f(x)\in\mathbb{Q}_p[x]$ such that $K\cong \mathbb{Q}_p(a)$, where $a$ is a root of $f(x)$.

The defining polynomial can be chosen to be monic with coefficients in $\mathbb{Z}_p$; by Krasner's lemma, we can further take $f(x)\in \mathbb{Z}[x]$.

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- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-24 16:34:53

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- 2020-10-24 16:34:53 by Andrew Sutherland (Reviewed)
- 2020-10-24 16:34:30 by Andrew Sutherland
- 2018-05-23 14:40:56 by John Cremona (Reviewed)

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