A User's Guide to Serre's Arithmetic/Hilbert Symbol

Local Properties edit

Definition and First Properties edit

For a fixed (local) field   the Hilbert symbol of two   is defined as


If we replace   by  , then


showing that if we multiply,   by squares, then their Hilbert symbols does not change. Hence the Hilbert symbol factors as


Serre goes on to prove that this is in fact a bilinear form over   in the next subsection.

After the definition he gives a method for computing the Hilbert Symbol in the proposition: It states that there is a short exact sequence


where   and


He then goes on to prove/state some identities useful for computation:

  7.   is proven in the theorem

Computation of   edit

Global Properties edit

Product Formula edit

Existence of Rational Numbers with given Hilbert Symbols edit

References edit

  1. https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec10.pdf