mathlib3 documentation


Separable degree #

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This file contains basics about the separable degree of a polynomial.

Main results #

Tags #

separable degree, degree, polynomial

def polynomial.is_separable_contraction {F : Type u_1} [comm_semiring F] (q : ) (f g : polynomial F) :

A separable contraction of a polynomial f is a separable polynomial g such that g(x^(q^m)) = f(x) for some m : ℕ.

def polynomial.has_separable_contraction {F : Type u_1} [comm_semiring F] (q : ) (f : polynomial F) :

The condition of having a separable contration.


A choice of a separable contraction.


The separable degree of a polynomial is the degree of a given separable contraction.


The separable degree divides the degree, in function of the exponential characteristic of F.

The separable degree divides the degree.

In exponential characteristic one, the separable degree equals the degree.

Every irreducible polynomial can be contracted to a separable polynomial.

theorem polynomial.contraction_degree_eq_or_insep {F : Type u_1} [field F] (q : ) [hq : ne_zero q] [char_p F q] (g g' : polynomial F) (m m' : ) (h_expand : (polynomial.expand F (q ^ m)) g = (polynomial.expand F (q ^ m')) g') (hg : g.separable) (hg' : g'.separable) :

If two expansions (along the positive characteristic) of two separable polynomials g and g' agree, then they have the same degree.

The separable degree equals the degree of any separable contraction, i.e., it is unique.