The **defining polynomial** of a local 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 $f(x)\in \mathbb{Z}[x]$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2018-05-23 14:40:56

**Referred to by:**

**History:**(expand/hide all)

- 2018-05-23 14:40:56 by John Cremona (Reviewed)