Much of this chapter is preparation for what we will do later
when we will prove that if is complete with respect to a valuation
(and locally compact) and is a finite extension of , then there
is a *unique* valuation on that extends the valuation on .
Also, if is a number field,
is a valuation on ,
is the completion of with respect to , and is a
finite extension of , we'll prove that

where the are the completions of with respect to the
equivalence classes of extensions of to . In particular,
if is a number field defined by a root of
, then

where the correspond to the irreducible factors of
the polynomial
(hence the extensions of
correspond to irreducible factors of
over
).
In preparation for this clean view of the local nature of number
fields, we will prove that the norms on a finite-dimensional
vector space over a complete field are all equivalent. We will also
explicitly construct tensor products of fields and deduce some of
their properties.

**Subsections**
William Stein
2012-09-24