If $K$ is a finite extension of $\mathbb{Q}_p$, then let $\mathcal{O}_K$ denote its ring of integers. Let $P$ be its unique maximal ideal. Then $\mathcal{O}_K/P$ is a field which is a finite degree extension of $\mathbb{Z}_p/p\mathbb{Z}_p = \mathbb{F}_p$. The residue field degree is the degree of this extension: $f = [\mathcal{O}_K/P : \mathbb{F}_p]$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2018-05-23 15:02:11

**Referred to by:**

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

- 2020-10-24 17:01:49 by Andrew Sutherland
- 2020-10-24 16:03:49 by Andrew Sutherland
- 2018-05-23 15:02:11 by John Cremona (Reviewed)

**Differences**(show/hide)