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 $K$ 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) $$\theta:K^*/{K^*}^n \to \mathrm{Gal}(L/K).$$ 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 [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 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 [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 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.

References

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

References

[a1]	K. Iwasawa, "Local class field theory" , Oxford Univ. Press (1986)
[a2]	J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. 4, Sect. 8