If $F$ is a finite extension of $\Q_p$ and $K$ a finite extension of $F$. Then $\mathcal{O}_F$ and $\mathcal{O}_K$, the ring of integers of $F$ and $K$ are discrete valuation domains, so they have unique maximal ideals $P_F$ and $P_K$ which are principal. If $P_F=(\pi_F)$, the element $\pi_F$ is a **uniformizer** for $F$.

The principal ideal $\pi_F\mathcal{O}_K=P_K^e$ for some positive integer $e$. The integer $e$ is the **ramification index** for $K$ over $F$. The ramification index of $K$ is then the ramification index for $K$ over $\Q_p$.

If $e=1$, then we say that the extension is **unramified**, and if $e=[K:\Q_p]$, then we say that the extension is **totally ramified**.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2018-05-30 08:52:33

**Referred to by:**

- lf.eisenstein_polynomial
- lf.invariants
- lf.unramified_degree
- lf.unramified_subfield
- lf.unramified_totally_ramified_tower
- lmfdb/local_fields/templates/lf-index.html (line 141)
- lmfdb/local_fields/templates/lf-search.html (line 21)
- lmfdb/local_fields/templates/lf-search.html (line 60)
- lmfdb/local_fields/templates/lf-show-field.html (line 19)
- lmfdb/motives/templates/motive-search.html (line 22)
- lmfdb/number_fields/templates/nf-show-field.html (line 242)

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

- 2018-05-30 08:52:33 by John Cremona (Reviewed)