Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Symbol (number theory)
Summary
In number theory, a symbol is any of many different generalizations of the Legendre symbol. This article describes the relations between these various generalizations.
The symbols below are arranged roughly in order of the date they were introduced, which is usually (but not always) in order of increasing generality.
Legendre symbol defined for p a prime, a an integer, and takes values 0, 1, or −1.
Jacobi symbol defined for b a positive odd integer, a an integer, and takes values 0, 1, or −1. An extension of the Legendre symbol to more general values of b.
Kronecker symbol defined for b any integer, a an integer, and takes values 0, 1, or −1. An extension of the Jacobi and Legendre symbols to more general values of b.
Power residue symbol is defined for a in some global field containing the mth roots of 1 ( for some m), b a fractional ideal of K built from prime ideals coprime to m. The symbol takes values in the m roots of 1. When m = 2 and the global field is the rationals this is more or less the same as the Jacobi symbol.
Hilbert symbol The local Hilbert symbol (a,b) = is defined for a and b in some local field containing the m roots of 1 (for some m) and takes values in the m roots of 1. The power residue symbol can be written in terms of the Hilbert symbol. The global Hilbert symbol is defined for a and b in some global field K, for p a finite or infinite place of K, and is equal to the local Hilbert symbol in the completion of K at the place p.
Artin symbol The local Artin symbol or norm residue symbol is defined for L a finite extension of the local field K, α an element of K, and takes values in the abelianization of the Galois group Gal(L/K). The global Artin symbol is defined for α in a ray class group or idele (class) group of a global field K, and takes values in the abelianization of Gal(L/K) for L an abelian extension of K. When α is in the idele group the symbol is sometimes called a Chevalley symbol or Artin–Chevalley symbol.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.