Lemma 9.7.7 (Multiplicativity). Suppose given a tower of fields F/E/k. Then
Proof. Let \alpha _1, \ldots , \alpha _ n \in F be an E-basis for F. Let \beta _1, \ldots , \beta _ m \in E be a k-basis for E. Then the claim is that the set of products \{ \alpha _ i \beta _ j, 1 \leq i \leq n, 1 \leq j \leq m\} is a k-basis for F. Indeed, let us check first that they span F over k.
By assumption, the \{ \alpha _ i\} span F over E. So if f \in F, there are a_ i \in E with
and, for each i, we can write a_ i = \sum b_{ij} \beta _ j for some b_{ij} \in k. Putting these together, we find
proving that the \{ \alpha _ i \beta _ j\} span F over k.
Suppose now that there existed a nontrivial relation
for the c_{ij} \in k. In that case, we would have
and the inner terms lie in E as the \beta _ j do. Now E-linear independence of the \{ \alpha _ i\} shows that the inner sums are all zero. Then k-linear independence of the \{ \beta _ j\} shows that the c_{ij} all vanish. \square
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