Projects and Assignments in Pure Mathematics
The p-adic numbers
The real numbers are obtained from the rationa numbers by `completing', i.e. adding limits of
Cauchy sequences, with respect to
Cauchy sequences. For any prime p, there is another `metric' or distance on the rational numbers. By completing with
respect to this, one obtains the field Q_p of p-adic numbers. This has
strange properties: for example, all
triangles are isosceles. The project could take various directions, for example the Hasse principle, that a quadratic form
over Q has a rational solution if and only if it has a solution in the reals and in all fields Q.
Another direction for the project would be to develop the general theory of valued fields.
Books
J.W.S. Cassels, Local fields, Cambridge Univ. Press, 1986.
P. Ribenboim, The theory of classical valuations, Springer, 1998.
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