Vector space over a field extension

Let be a field extension and be an -vector space. Then is also a -vector space, and #m/thm/linalg

Proof

That is a vector space over any subfield of , so in particular it is an -vector space. Let be an -indexed -basis for and be a -indexed -basis for . We claim that

forms an -indexed -basis for . Indeed, for any , we have

for some finite subset , and for each we have

for some finite subset . Therefore

is a finite linear combination.

Corollaries

Intermediate field extension

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