Nice proof of the Primitive Element Theorem for infinite base fields (though it does rely on having done stuff in a different order to Yoshida). We will assume two facts, and that K is any infinite field.

Fact 1: Every separable extension has only finitely many subextensions.

Fact 2: No finite-dimensional vector space over an infinite field is a union of finitely many of its proper subspaces.

Theorem (Primitive Element Theorem): If F/K is finite and separable then it has a primitive element (\theta such that F=K(\theta)).

Proof: Let X = F \backslash (\bigcup_{K\leq M < F} M). By fact 1, this is a finite union and so by fact 2, X is nonempty. Hence any element of X is a primitive element.

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