Let R=C(u,v) be a subfield of C(x,y) with u,v in C[x,y] algebraically independent over C, with u not a square in R.
Take L=R(s), where s^2=u, so [L:R]=2.
Clearly, L is a subfield of an algebraic closure of C(x,y).
Please see the following MathStackExchange question of mine: https://math.stackexchange.com/questions/4942855/constructing-an-automorphism-between-mathbbcx-y-and-l-subseteq-overlin
Shortly, I wish to show that under certain conditions, [C(x,y):R]=2.
Thank you very much!
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