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 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 ( such that ).

**Proof: **Let . By fact 1, this is a finite union and so by fact 2, is nonempty. Hence any element of is a primitive element.

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July 30, 2010 at 4:46 pm

astrohockeyThis is nice. The blogosphere has missed you.