# Conductor of a character

An integer associated to the character of a representation of the Galois group of a finite extension of a local field. Let be a field that is complete with respect to a discrete valuation, with residue class field of characteristic . Let be a Galois extension of degree with Galois group and suppose that the residue class field extension is separable. If is the character of some finite-dimensional complex representation of , its conductor is defined by the formula:

where

where is the corresponding valuation of . If does not divide , then for and . If is the character of a rational representation , then . The conductor is a non-negative integer.

#### References

[1] | J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1967) pp. Chapt. VI |

[2] | E. Artin, J. Tate, "Class field theory" , Benjamin (1967) |

[3] | J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) |

#### Comments

The ideal , where is the conductor of a character of the Galois group of an extension of local fields, is also called the Artin conductor of . There is a corresponding notion for extensions of global fields obtained by taking a suitable product over all finite primes, cf. [a1], p. 126. It plays an important role in the theory of Artin -functions, cf. -function.

#### References

[a1] | J. Neukirch, "Class field theory" , Springer (1986) |

**How to Cite This Entry:**

Conductor of a character. I.V. Dolgachev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Conductor_of_a_character&oldid=19015