Normed Spaces and Tensor Products

Much of this chapter is preparation for what we will do later when we will prove that if $ K$ is complete with respect to a valuation (and locally compact) and $ L$ is a finite extension of $ K$, then there is a unique valuation on $ L$ that extends the valuation on $ K$. Also, if $ K$ is a number field, $ v=\left\vert \cdot \right\vert{}$ is a valuation on $ K$, $ K_v$ is the completion of $ K$ with respect to $ v$, and $ L$ is a finite extension of $ K$, we'll prove that

$\displaystyle K_v \otimes _K L = \bigoplus_{j=1}^J L_j,

where the $ L_j$ are the completions of $ L$ with respect to the equivalence classes of extensions of $ v$ to $ L$. In particular, if $ L$ is a number field defined by a root of $ f(x)\in\mathbf{Q}[x]$, then

$\displaystyle \mathbf{Q}_p \otimes _\mathbf{Q}L = \bigoplus_{j=1}^J L_j,

where the $ L_j$ correspond to the irreducible factors of the polynomial $ f(x) \in \mathbf{Q}_p[x]$ (hence the extensions of $ \left\vert \cdot \right\vert _p$ correspond to irreducible factors of $ f(x)$ over $ \mathbf{Q}_p[x]$).

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.

William Stein 2004-05-06