Hilbert symbol
In
The Hilbert symbol has been generalized to higher local fields.
Quadratic Hilbert symbol
Over a local field K whose multiplicative group of non-zero elements is K×, the quadratic Hilbert symbol is the function (–, –) from K× × K× to {−1,1} defined by
Equivalently, if and only if is equal to the norm of an element of the quadratic extension [1] page 110.
Properties
The following three properties follow directly from the definition, by choosing suitable solutions of the diophantine equation above:
- If a is a square, then (a, b) = 1 for all b.
- For all a,b in K×, (a, b) = (b, a).
- For any a in K× such that a−1 is also in K×, we have (a, 1−a) = 1.
The (bi)multiplicativity, i.e.,
- (a, b1b2) = (a, b1)·(a, b2)
for any a, b1 and b2 in K× is, however, more difficult to prove, and requires the development of local class field theory.
The third property shows that the Hilbert symbol is an example of a Steinberg symbol and thus factors over the second Milnor K-group , which is by definition
- K× ⊗ K× / (a ⊗ (1−a), a ∈ K× \ {1})
By the first property it even factors over . This is the first step towards the Milnor conjecture.
Interpretation as an algebra
The Hilbert symbol can also be used to denote the central simple algebra over K with basis 1,i,j,k and multiplication rules , , . In this case the algebra represents an element of order 2 in the Brauer group of K, which is identified with -1 if it is a division algebra and +1 if it is isomorphic to the algebra of 2 by 2 matrices.
Hilbert symbols over the rationals
For a
Over the reals, (a, b)∞ is +1 if at least one of a or b is positive, and −1 if both are negative.
Over the p-adics with p odd, writing and , where u and v are integers
- , where
and the expression involves two Legendre symbols.
Over the 2-adics, again writing and , where u and v are
- , where
It is known that if v ranges over all places, (a, b)v is 1 for almost all places. Therefore, the following product formula
makes sense. It is equivalent to the law of quadratic reciprocity.
Kaplansky radical
The Hilbert symbol on a field F defines a map
where Br(F) is the Brauer group of F. The kernel of this mapping, the elements a such that (a,b)=1 for all b, is the Kaplansky radical of F.[2]
The radical is a subgroup of F*/F*2, identified with a subgroup of F*. The radical is equal to F* if and only if F has u-invariant at most 2.[3] In the opposite direction, a field with radical F*2 is termed a Hilbert field.[4]
The general Hilbert symbol
If K is a local field containing the group of nth roots of unity for some positive integer n prime to the characteristic of K, then the Hilbert symbol (,) is a function from K*×K* to μn. In terms of the Artin symbol it can be defined by[5]
Hilbert originally defined the Hilbert symbol before the Artin symbol was discovered, and his definition (for n prime) used the power residue symbol when K has residue characteristic coprime to n, and was rather complicated when K has residue characteristic dividing n.
Properties
The Hilbert symbol is (multiplicatively) bilinear:
- (ab,c) = (a,c)(b,c)
- (a,bc) = (a,b)(a,c)
skew symmetric:
- (a,b) = (b,a)−1
nondegenerate:
- (a,b)=1 for all b if and only if a is in K*n
It detects norms (hence the name norm residue symbol):
- (a,b)=1 if and only if a is a norm of an element in K(n√b)
It has the "symbol" properties:
- (a,1–a)=1, (a,–a)=1.
Hilbert's reciprocity law
Hilbert's reciprocity law states that if a and b are in an algebraic number field containing the nth roots of unity then[6]
where the product is over the finite and infinite primes p of the number field, and where (,)p is the Hilbert symbol of the completion at p. Hilbert's reciprocity law follows from the
Power residue symbol
If K is a number field containing the nth roots of unity, p is a prime ideal not dividing n, π is a prime element of the local field of p, and a is coprime to p, then the power residue symbol (a
p) is related to the Hilbert symbol by[7]
The power residue symbol is extended to fractional ideals by multiplicativity, and defined for elements of the number field
by putting (a
b)=(a
(b)) where (b) is the principal ideal generated by b.
Hilbert's reciprocity law then implies the following reciprocity law for the residue symbol, for a and b prime to each other and to n:
See also
External links
- "Norm-residue symbol", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- HilbertSymbol at Mathworld
References
- Zbl 0145.04902
- ISSN 0012-0456
- MR 1646901
- Zbl 1068.11023
- Zbl 0237.18005
- Zbl 0956.11021
- Zbl 0256.12001
- Vostokov, S. V.; Fesenko, I. B. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121, Providence, R.I.: Zbl 1156.11046