Dirichlet character
A Dirichlet character modulo is a group homomorphism from to . Dirichlet characters are usually denoted by the Greek letter . The function
is also referred to as a Dirichlet character. The Dirichlet characters modulo form a group if one defines to be the function which takes to . It turns out that this resulting group is isomorphic to . The trivial character is given by for all , and it acts as the identity element for the group. A character modulo is said to be induced by a character modulo if and . A character which is not induced by any other character is called primitive. If is non-primitive, the of all such is called the conductor of .
Examples:
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Legendre symbol is a Dirichlet character modulo for any odd prime . More generally, Jacobi symbol is a Dirichlet character modulo .
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The character modulo given by and is a primitive character modulo . The only other character modulo is the trivial character.
Title | Dirichlet character |
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Canonical name | DirichletCharacter |
Date of creation | 2013-03-22 13:22:31 |
Last modified on | 2013-03-22 13:22:31 |
Owner | bbukh (348) |
Last modified by | bbukh (348) |
Numerical id | 10 |
Author | bbukh (348) |
Entry type | Definition |
Classification | msc 11A25 |
Related topic | CharacterOfAFiniteGroup |
Defines | trivial character |
Defines | primitive character |
Defines | conductor |
Defines | induced character |