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$E/K$ is a field extension. Suppose $\exists m\in Z_{>0}, \forall x\in E, [K(x):K]\leq m$.

$\textbf{Q1:}$ Do I need separability to deduce that $E/K$ is a finite extension of degree at most $m$?(I did use separability to deduce simple extension and this makes my life significantly easier as I just run the argument of simple extension.)

$\textbf{Q2:}$ If I do need separability to deduce $E/K$ finite extension, please give a non-finite extension with every element having minimal polynomial with uniform bound on the degree of minimal polynomial. It is clear that counter example should come from $char\neq 0$ case.

user45765
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1 Answers1

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After only a few minutes’ thought, I did not see an example in characteristic zero. But let $k=\Bbb F_p$, $K=k(t_1,t_2,\cdots)$, what you get by adjoining infinitely many independent transcendental elements to $k$. Now let $E=K^p$, the set of all $p$-th powers of elements of $K$. Then each $x\in K$ satisfies $x^p\in E$, so every simple extension has field extension degree $p$. Of course the total extension is not finite-dimensional.

Lubin
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