Re: Finite field extensions

On 1 Mar, 15:00, "Grothendieck-Hirzebruch" <Grothendieck-
Hirzebr...@xxxxxx> wrote:
If H and H' are subgroups of finite index of a group G, then their
intersection H/cap H' again is a subgroup of finite index. I'm
wondering if an analogous statement is true for fields:
So let L be a field, K_1 and K_2 subfields such that L is a finite-
dimensional K_i-vector space for i=1,2. Let k:=K_1 \cap K_2 be the
intersection of the fields - is L then a finite-dimensional k-vector
space?

No.

As an example take L = C, the complex numbers and K_1 = R, the
real numbers. As a consequence of Zorn's lemma, C has uncountably
many automorphisms. Let sigma be one of these but not the
identity or complex conjugation. Let K_2 = sigma(R). Then K_2
is not equal to R. We cannot have C finite over R intersect K_2
for then its degree would be at least 4, and if an algebraically
closed field is a finite extension of a subfield, its degree
is either 1 or 2 (a theorem of Artin and Schreier).

Victor Meldrew

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