# 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 $F$.