TOPICS
Search

Hilbert Symbol


For any two nonzero p-adic numbers a and b, the Hilbert symbol is defined as

 (a,b)={1   if z^2=ax^2+by^2 has a nonzero solution; -1   otherwise.
(1)

If the p-adic field is not clear, it is said to be the Hilbert symbol of a and b relative to k. The field can also be the reals (p=infty). The Hilbert symbol satisfies the following formulas:

1. (a,b)=(b,a).

2. (a,c^2)=1 for any c.

3. (a,-a)=1.

4. (a,1-a)=1.

5. (a,b)=1=>(aa^',b)=(a^',b).

6. (a,b)=(a,-ab)=(a,(1-a)b).

The Hilbert symbol depends only the values of a and b modulo squares. So the symbol is a map k^*/k^*^2×k^*/k^*^2->{1,-1}.

Hilbert showed that for any two nonzero rational numbers a and b,

1. (a,b)_v=1 for almost every prime v.

2. product(a,b)_v=1 where v ranges over every prime, including v=infty corresponding to the reals.


See also

Diophantine Equation--2nd Powers, Field, p-adic Number, Symmetric Bilinear Form, Vector Space

This entry contributed by Todd Rowland

Explore with Wolfram|Alpha

Cite this as:

Rowland, Todd. "Hilbert Symbol." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HilbertSymbol.html

Subject classifications