---
tag: B4VK33PCXZ
kind: concept
name: Regular prime
msc: [11R18, 11R29]
specializes: [01826KMG2H]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_prime
same_as:
  wikidata: Q426491
  wikipedia: Regular prime
---

## Summary

An odd [[prime number|01826KMG2H]] $p$ is regular if it does not divide the [[class number|KTKK9B1X41]] of the [[cyclotomic field|NN77VYSGXR]] $\Q(\zeta_p)$.
Kummer introduced regular primes and proved [[Fermat's Last Theorem|0Y5X3SDBCZ]] for every regular prime exponent.
By Kummer's criterion, $p$ is regular exactly when $p$ divides none of the numerators of the [[Bernoulli numbers|0KC00K7KY4]] $B_2, B_4, \ldots, B_{p-3}$.
The smallest irregular prime is $37$.

## Explanations

### L1 human

Kummer tried to prove Fermat's Last Theorem by factoring $x^p + y^p$ in the ring generated by a $p$-th root of unity.
That ring does not always have unique factorisation, and the class number measures the failure.
When $p$ does not divide this class number, the failure is mild enough for his argument to work; such primes are called regular.
All odd primes below $37$ are regular, while $37$ is not, because $37$ divides the numerator of $B_{32}$.