View source for Norm-residue symbol

''norm residue, Hilbert symbol''

A function that associates with an ordered pair of elements $x,y$ of the
multiplicative group $K^*$ of a
[[Local field|local field]] $K$ an element $(x,y)\in K^*$ that is an $n$-th root
of unity. This function can be defined as follows. Let $\zeta_n\in K$ be a
primitive $n$-th root of unity. The maximal Abelian extension $L$ of
$K$ with Galois group $G(L/K)$ of exponent $n$ is obtained by adjoining to
$K$ the roots $a^{1/n}$ for all $a\in K^*$. On the other hand, there is a canonical
isomorphism (the fundamental isomorphism of local
[[Class field theory|class field theory]]) 
$$\theta:K^*/{K^*}^n \to \mathrm{Gal}(L/K).$$
The norm residue of
the pair $(x,y)$ is defined by 
$$\theta(y)(x^{1/n}) = (x,y)x^{1/n}.$$
D. Hilbert introduced the concept of
a norm-residue symbol in the special case of quadratic fields with
$n=2$. In
[[#References|[4]]] there is an explicit definition of the norm
residue using only local class field theory.

Properties of the symbol $(x,y)$:

1) bilinearity: $(x_1x_2,y) = (x_1,y)(x_2,y),\quad (x,y_1y_2)=(x,y_1)(x,y_2)$;

2) skew-symmetry: $(x,y)(y,x)=1$;

3) non-degeneracy: $(x,y)=1$ for all $x\in K^*$ implies $y\in {K^*}^n$; $(x,y)=1$ for all $y\in K^*$
implies $x\in {K^*}^n$;

4) if $x+y=1$, then $(x,y)=1$;

5) if $\sigma$ is an automorphism of $K$, then 
$$(\sigma x, \sigma y) = \sigma(x,y)$$
6) let $K'$ be a finite
extension of $K$, $a\in {K'}^*$
and $b\in K^*$. Then 
$$(a,b) = (N_{K'/K}(a),b)$$
where on the left-hand side
the norm-residue symbol is regarded for $K'$ and on the right-hand side
that for $K$, and where $N_{K'/K}$ is the
[[Norm map|norm map]] from $K'$ into $K$;

7) $(x,y)=1$ implies that $y$ is a norm in the extension $K(x^{1/n})$. (This explains
the name of the symbol.)

The function $(x,y)$ induces a non-degenerate bilinear pairing 
$$K^*/{K^*}^n \times K^*/{K^*}^n \to \mu(n)$$
where
$\mu(n)$ is the group of roots of unity generated by $\zeta_n$. Let $\Psi:K^*\times K^* \to A$ be a
mapping into some Abelian group $A$ satisfying 1), 4) and the
condition of continuity: For any $y\in K^*$ the set $\{x\in K^* | \Psi(x,y)=1\}$ is closed in $K^*$. The
norm-residue symbol has the following universal property
[[#References|[3]]]: If $n$ is the number of roots of unity in $K$,
then there exists a homomorphism $\phi:\mu(n) \to A$ such that for any $x,y\in K^*$, 
$$\Psi(x,y) = \phi((x,y)).$$
This
property can serve as a basic axiomatic definition of the norm-residue
symbol.

If $F$ is a
[[Global field|global field]] and $K$ is the completion of $F$
relative to a place $\nu$, then by the norm-residue symbol one also
means the function $(x,y)_\nu$ defined over $F^*\times F^*$ that is obtained by
composition of the (local) norm-residue symbol $(x,y)$ with the natural
imbedding $F^*\to K^*$.

Often the norm-residue symbol is defined as an automorphism $\theta(x)$ of the
maximal Abelian extension of $K$ corresponding to an element $x\in K^*$ by
local
[[Class field theory|class field theory]].

====References====
<table><TR><TD valign="top">[1]</TD>
 <TD valign="top"> J.W.S. Cassels (ed.)  A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986)</TD>
</TR><TR><TD valign="top">[2]</TD>
 <TD valign="top"> H. Koch, "Galoissche Theorie der $p$-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)</TD>
</TR><TR><TD valign="top">[3]</TD>
 <TD valign="top"> J.W. Milnor, "Introduction to algebraic $K$-theory" , Princeton Univ. Press (1971)</TD>
</TR><TR><TD valign="top">[4]</TD>
 <TD valign="top"> I.R. Shafarevich, "A general reciprocity law" ''Mat. Sb.'' , '''26''' : 1 (1950) pp. 113–146 (In Russian)</TD>
</TR></table>

====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD
valign="top"> K. Iwasawa, "Local class field theory" , Oxford
Univ. Press (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD
valign="top"> J. Neukirch, "Class field theory" , Springer (1986)
pp. Chapt. 4, Sect. 8</TD></TR></table>