The **Artin field** is a number field associated to an Artin representation
$\rho:\mathrm{Gal}(\overline{\Q}/\Q)\to\GL(V)$
by being the smallest Galois extension $K/\mathbb{Q}$ such that $\rho$ factors through
$\mathrm{Gal}(K/\Q)$.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2015-08-11 17:01:16

**Referred to by:**

- artin
- artin.gg_quotient
- artin.ramified_primes
- artin.stem_field
- artin.unramified_primes
- lmfdb/artin_representations/main.py (line 773)
- lmfdb/artin_representations/main.py (line 788)
- lmfdb/artin_representations/templates/artin-representation-galois-orbit.html (line 16)
- lmfdb/artin_representations/templates/artin-representation-set-show.html (line 10)
- lmfdb/artin_representations/templates/artin-representation-show.html (line 19)

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

- 2015-08-11 17:01:16 by John Cremona (Reviewed)