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When designing friendly problems for a calculus class one comes up with such a question. (The cubic case is relatively easy.)

Of course one can generalize: Characterize degree $n$ polynomials such that the first $k$ derivatives, of order $0$ through $k-1$, have roots only in a certain field.

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    Won't it just give some ugly conditions on the coefficients, even in the cubic case? Or is there some natural way to look at this? I didn't give it some thought but I wondered if this is natural in some context, it feels like it could.2012-10-01
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    I changed tags ; it's not because we use field theory in number theory that the number theory tag is appropriate.2012-10-01

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See the paper, W. P. Galvin and J. A. MacDougall,`Nice' Quartic Polynomials - The Sequel.