# Norm-residue symbol

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norm residue, Hilbert symbol

A function that associates with an ordered pair of elements of the multiplicative group of a local field an element that is an -th root of unity. This function can be defined as follows. Let be a primitive -th root of unity. The maximal Abelian extension of with Galois group of exponent is obtained by adjoining to the roots for all . On the other hand, there is a canonical isomorphism (the fundamental isomorphism of local class field theory) The norm residue of the pair is defined by D. Hilbert introduced the concept of a norm-residue symbol in the special case of quadratic fields with . In  there is an explicit definition of the norm residue using only local class field theory.

Properties of the symbol :

1) bilinearity: , ;

2) skew-symmetry: ;

3) non-degeneracy: for all implies ; for all implies ;

4) if , then ;

5) if is an automorphism of , then 6) let be a finite extension of , and . Then where on the left-hand side the norm-residue symbol is regarded for and on the right-hand side that for , and where is the norm map from into ;

7) implies that is a norm in the extension . (This explains the name of the symbol.)

The function induces a non-degenerate bilinear pairing where is the group of roots of unity generated by . Let be a mapping into some Abelian group satisfying 1), 4) and the condition of continuity: For any the set is closed in . The norm-residue symbol has the following universal property : If is the number of roots of unity in , then there exists a homomorphism such that for any , This property can serve as a basic axiomatic definition of the norm-residue symbol.

If is a global field and is the completion of relative to a place , then by the norm-residue symbol one also means the function defined over that is obtained by composition of the (local) norm-residue symbol with the natural imbedding .

Often the norm-residue symbol is defined as an automorphism of the maximal Abelian extension of corresponding to an element by local class field theory.

How to Cite This Entry:
Norm-residue symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Norm-residue_symbol&oldid=12271
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article