Every intermediate field of an infinite Galois extension is the union of finite extensions I was stumbling upon this statement in my study of infinite Galois extensions but it had no further explanation. It seems true to me but I don't know how to construct these finite extensions. Thanks for the help in advance!

Well, not very surprisingly, this is true for any algebraic extension. If $ L/K $ is an algebraic extension of fields, then $ L = \cup_{x \in L} K(x) $, and each $ K(x) $ is finite over $ K $ since $ x \in L $, and $ L $ is algebraic over $ K $.