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Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?

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Seems highly unlikely to me. Here's a possible strategy for a counterexample. Let $K$ be a $p$-adic field and let $L$ be its maximal tamely ramified extension. Then $\Gamma:=Gal(L/K)$ is a well-understood group and it's not abelian. Find two elements that don't commute and finite extensions $M/N/K$ in $L$ such that the elements fix $N$ pointwise, and restrict to commuting elements of $Gal(M/N)$ with the property that no lifts of these elements to $Gal(L/N)$ commute. This is now just a messy problem in group theory. Then $M$ and the automorphisms, if you can find them, give a counterexample. – Kevin Buzzard May 27 '12 at 20:08
Why do we even expect the endomorphisms to extend at all to the algebraic closure? – J. Martel May 28 '12 at 20:14
In your setting, an appropriate justification for the existence of an extension should use your hypotheses that the field is complete and discrete valued. – J. Martel May 28 '12 at 20:16

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