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

Local PropertiesEdit

Definition and First PropertiesEdit

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

  sends  

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

  1.  
  2.  
  3.  
  4.  
  5.  
  6.  
  7.   is proven in the theorem

Computation of  Edit

Global PropertiesEdit

Product FormulaEdit

Existence of Rational Numbers with given Hilbert SymbolsEdit

ReferencesEdit

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