Number Theory Lecture 6

This lecture (NumTh6) was given on 9/17/09

Excerpts

A null sequence is a sequence converging to 0.  Given a fields F with an absolute value |-|.  Its completetion (all cauchy sequence modulo null sequences) is denoted $lates \hat F$.

Results:

  1. For archimedean fiels F \subset \mathbb{C}, \hat F = \mathbb{R} or \mathbb{C}.
  2. For a non archimedean field F, the set of all values |\hat F |\subset \mathbb{R} obtained as absolute values of elements in \hat F is the same as |F|, the set of all values obtained as absolute values of elements in F.
  3. With F as in 2, the residue field (def in prev. lecture) of F is the same as that of \hat F.

(F, |-|) is a local field if its non archimedean, the absolute value is induced by a discrete valuation, F is complete, and its residue field \mathbb{F} is finite.

Note sitting inside F is a DVR R.  Using a uniformizing parameter t, any nonzero element can be expressed as

\alpha = \sum_{n=e}^\infty a_n t^n

for e \in \mathbb{Z}.  This allows one to give a description of the topology on R and F.

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