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Consider the following statement: "Let $L/K$ be an algebraic field extension. Then the maximal separable sub-extension is finite."

Here is what seems to be a proof: "Let $M/K$, $K \subset M \subset L$ be a finite separable subextension of maximal degree. Let $x \in L$ be separable over $K$. So it will be separable over $M$. Consider the field tower $K \subset M \subset M(x)$. Each step of this tower is separable and finite and so $M(x) / K$ is finite and separable. But then $x \in M$ by the maximality assumption. This shows that $M/K$ is in fact the maximal separable sub-extension."

Could you confirm the validity of the above proof? Thanks.

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    $\begingroup$ The quoted result is false: take for instance $K = \mathbb{Q}$ and $L = \overline{\mathbb{Q}}$. (And the very first step of the proof is faulty: there need not be a finite separable subextension of maximal degree!) $\endgroup$ – Pete L. Clark Aug 8 '12 at 14:36
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The quoted result is false: take for instance $K = \mathbb{Q}$ and $L = \overline{\mathbb{Q}}$.

The first step of the purported proof is faulty: evidently there need not be a finite separable subextension of maximal degree!

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