A **prime** $\mathfrak p$ of a number field $K$ is a nonzero prime ideal of its ring of integers $\mathcal O_K$.

The ideal $\mathfrak p \cap\mathcal O_K$ is a nonzero prime ideal of $\Z$ (a prime of $\Q$), which is necessarily a principal ideal $(p)$ for some prime number $p$. The prime $\mathfrak p$ is then said to be a **prime above** $p$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2019-10-29 15:26:52

**Referred to by:**

- ag.good_reduction
- ag.potential_good_reduction
- artin.conductor
- av.potential_toric_rank
- ec.bad_reduction
- ec.bsdconjecture
- ec.good_reduction
- ec.local_root_number
- modlgal.conductor
- modlgal.frobenius_charpoly
- modlgal.frobenius_order
- modlgal.frobenius_trace
- modlgal.ramified
- nf.padic_completion
- rcs.source.ec
- st_group.weight

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

**Differences**(show/hide)