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

Local Properties Definition and First Properties 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: is proven in the theorem Computation of Global Properties Product Formula Existence of Rational Numbers with given Hilbert Symbols References https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec10.pdf