# 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 with an absolute value . Its completetion (all cauchy sequence modulo null sequences) is denoted $lates \hat F$.

Results:

- For archimedean fiels , or .
- For a non archimedean field , the set of all values obtained as absolute values of elements in is the same as , the set of all values obtained as absolute values of elements in .
- With as in 2, the residue field (def in prev. lecture) of is the same as that of .

is a local field if its non archimedean, the absolute value is induced by a discrete valuation, is complete, and its residue field is finite.

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

for . This allows one to give a description of the topology on and .

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- Published:
- September 20, 2009 / 11:30 pm

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- Number Theory (course)

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