Are there some necessary conditions on the coefficients of a monic, irreducible polynomial $\Phi$ over a local field?
irreducible polynomials over the $p$ adic number
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number-theory
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4The constant term must be non-zero. – 2011-12-19
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7If $f$ is irreducible, then the newton polygon of $f$ has exactly one slope. – 2011-12-19
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4The first comment is not quite right: try x. :) – 2011-12-20
1 Answers
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A corollary of Hensels lemma: If K is a field that is complete with respect to a non-archimedean absolute value, then for an irreducible polynomial $P=\sum{a_iX^i}$ of degree $n$ we have max($|a_i|$)=max($|a_0|$,$|a_n|$)
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0Perfect answer: Actually, I was hoping exactly for a statement of this flavor. – 2011-12-20
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1Corollary 4.7 in Chapter 2 of Neukirch "Algebraic number theory". – 2011-12-20