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

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

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

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

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.