A prime integer $p$ is a **ramified prime** of a number field $K$ if, when the ideal generated by $p$ is factored into prime ideals in the ring of integers
$\mathcal{O}_K$ of $K$,
$$p\mathcal{O}_K= \mathcal{P_1}^{e_1}\cdots \mathcal{P_k}^{e_k},$$

there is an $i$ such that $e_i\geq 2$.

The ramified primes of $K$ are the primes dividing the discriminant of $K$.

**Knowl status:**

- Review status: reviewed
- Last edited by John Jones on 2018-03-20 18:02:02

**Referred to by:**

- artin.ramified_primes
- nf.discriminant
- nf.ideal_labels
- nf.invariants
- lmfdb/number_fields/templates/number_field.html (line 36)
- lmfdb/number_fields/templates/number_field.html (lines 230-235)
- lmfdb/number_fields/templates/number_field_all.html (line 102)
- lmfdb/number_fields/templates/number_field_search.html (line 27)

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

- 2018-03-20 18:02:02 by John Jones (Reviewed)