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.
|||J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)|
|||H. Koch, "Galoissche Theorie der -Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)|
|||J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)|
|||I.R. Shafarevich, "A general reciprocity law" Mat. Sb. , 26 : 1 (1950) pp. 113–146 (In Russian)|
|[a1]||K. Iwasawa, "Local class field theory" , Oxford Univ. Press (1986)|
|[a2]||J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8|
Norm-residue symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Norm-residue_symbol&oldid=12271