5
$\begingroup$

Are there some necessary conditions on the coefficients of a monic, irreducible polynomial $\Phi$ over a local field?

  • 4
    The constant term must be non-zero.2011-12-19
  • 7
    If $f$ is irreducible, then the newton polygon of $f$ has exactly one slope.2011-12-19
  • 4
    The first comment is not quite right: try x. :)2011-12-20

1 Answers 1

7

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($|a_0|$,$|a_n|$)

  • 0
    Perfect answer: Actually, I was hoping exactly for a statement of this flavor.2011-12-20
  • 1
    Corollary 4.7 in Chapter 2 of Neukirch "Algebraic number theory".2011-12-20