# Local Theory

1. Define a valuation on a field. Characterize archimedean and non-archimedean valuations. What are equivalent valuations?

2. What is Ostrowski's theorem?

3. Describe the -adic numbers and integers. Characterize the -adic numbers as Laurent series in . Describe the -adic integers in terms of Laurent series in and in terms of the -adic valuation on . Show that is a discrete valuation ring. Characterize all the ideals of .

4. Determine all the archimedean valuations on .

5. For an extension of number fields and a prime in , describe the normalized valuation on . Describe all finite extensions of , their valuations, and degrees.

6. Let be an odd prime in . Use Hensel's lemma to prove there are precisely three quadratic extensions of .

root 2007-06-06