Separable degree #
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This file contains basics about the separable degree of a polynomial.
Main results #
is_separable_contraction: is the condition that, for
ga separable polynomial, we have that
g(x^(q^m)) = f(x)for some
m : ℕ.
has_separable_contraction: the condition of having a separable contraction
has_separable_contraction.degree: the separable degree, defined as the degree of some separable contraction
irreducible.has_separable_contraction: any irreducible polynomial can be contracted to a separable polynomial
has_separable_contraction.dvd_degree': the degree of a separable contraction divides the degree, in function of the exponential characteristic of the field
has_separable_contraction.eq_degreespecialize the statement of
is_separable_contraction.degree_eq: the separable degree is well-defined, implemented as the statement that the degree of any separable contraction equals
separable degree, degree, polynomial
A separable contraction of a polynomial
f is a separable polynomial
g such that
g(x^(q^m)) = f(x) for some
m : ℕ.
The condition of having a separable contration.
- polynomial.has_separable_contraction q f = ∃ (g : polynomial F), polynomial.is_separable_contraction q f g
A choice of a separable contraction.
- hf.contraction = classical.some hf
The separable degree of a polynomial is the degree of a given separable contraction.
- hf.degree = hf.contraction.nat_degree
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. https://stacks.math.columbia.edu/tag/09H0
If two expansions (along the positive characteristic) of two separable polynomials
agree, then they have the same degree.
The separable degree equals the degree of any separable contraction, i.e., it is unique.