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 $ p$-adic numbers and integers. Characterize the $ p$-adic numbers as Laurent series in $ p$. Describe the $ p$-adic integers in terms of Laurent series in $ p$ and in terms of the $ p$-adic valuation on $ {\mathbb{Q}}_p$. Show that $ {\mathbb{Z}}_p$ is a discrete valuation ring. Characterize all the ideals of $ {\mathbb{Z}}_p$.

  4. Determine all the archimedean valuations on $ {\mathbb{Q}}(\root 3 \of 2)$.

  5. For an extension of number fields $ L/K$ and $ {\mathfrak{p}}$ a prime in $ \O _K$, describe the normalized valuation $ \vert\cdot \vert _{\mathfrak{p}}$ on $ K$. Describe all finite extensions of $ K_{\mathfrak{p}}$, their valuations, and degrees.

  6. Let $ p$ be an odd prime in $ {\mathbb{Z}}$. Use Hensel's lemma to prove there are precisely three quadratic extensions of $ {\mathbb{Q}}_p$.

root 2007-06-06