Abstract

We will give a very elementary introduction to reciprocity laws ( of which the Langlands programme is a vast generalisation). Starting with the definition of the field Q_p of p-adic numbers, we will prove that the Galois group G of the compositum M of all quadratic extensions of Q_p is canonically isomorphic to the multiplicative group modulo squares. We also give an explicit formula for the quadratic Hilbert symbol, which gives the action of G on M.
The talks will be accessible to a very wide audience and will be adapted to the background and the interests of the audience..



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