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In an application of Krasner's lemma, to show that the algebraic closure $\theta$ of $Q_p$ is not topologically complete, one shows that the dimension of $\theta$ is at most infinitely countable over $Q_p$, and so by Blaire category theorem we finish.

For this, I need the following result:

If we consider ${Q_p}^{n+1}$ as the space of polynomial of degree at most $n$ (in the obvious way), then if we have a separable irreducible polynomial $f$, then if we slightly perturb its coefficients, it remains irreducible.

This isn't hard once I can prove my main question:

If we perturb the coefficients, then the roots also are pertrubed slightly.

How can one show this?

Andy
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  • Would you add to the body of your Question a specification of what the notation $Q_p$ means? Indeed you have not said anything regarding $p$ apart from its use in this notation. – hardmath Sep 19 '18 at 17:19
  • @hardmath It's the p-adics, i.e the compleition of $Q$ with respect to the $p$ metric (for some prime $p$), isn't this standard notation? – Andy Sep 19 '18 at 17:20
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    Related: https://math.stackexchange.com/questions/63196/continuity-of-the-roots-of-a-polynomial-in-terms-of-its-coefficients – rogerl Sep 19 '18 at 18:05
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    I understood the notation $Q_p$ from the context and the p-adic-number-theory tag; maybe even more standard is $\Bbb Q_p$, which you can produce with \Bbb Q_p. -- As for the actual question, Pete L. Clark's answer to the question linked by @rogerl gives a reference for this exact problem. – Torsten Schoeneberg Sep 19 '18 at 19:08
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    Consider the map $(\rho_1,\cdots,\rho_n)\mapsto(S_1,\cdots,S_n)$ where the $S_i$ are the coefficients of the polynomial whose roots are the $\rho_i$. I think that you’ll find that the square of the determinant of the ($n\times n$) Jacobian matrix is the discriminant of the polynomial. Thus when, as here, the roots are distinct, the map has a local inverse. – Lubin Sep 20 '18 at 04:08
  • @Lubin Do you have a reference to such a theorem over the p-adics? Or maybe it's the same as in $R^n$ – Andy Sep 20 '18 at 11:40
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    It’s not a $p$-adic theorem, it’s plain sit-down-and-prove-it algebra. I once wrote out a proof, but that was in the days before electrons were discovered, so I have no remnant of it. I’ll think about reconstructing a proof. (It has to be a well-known fact.) Why don’t you e-mail me so I can maintain contact? – Lubin Sep 20 '18 at 15:09
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    @Lubin Thanks, but if it's that simple I think I'd rather try to prove it myself :) – Andy Sep 20 '18 at 21:59

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