# Normed Spaces and Tensor Products

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