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

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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:

  1. is proven in the theorem

Computation of [edit]

Global Properties[edit]

Product Formula[edit]

Existence of Rational Numbers with given Hilbert Symbols[edit]


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