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ring_theory.polynomial.eisenstein.basic

Eisenstein polynomials #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4. Given an ideal π“Ÿ of a commutative semiring R, we say that a polynomial f : R[X] is Eisenstein at π“Ÿ if f.leading_coeff βˆ‰ π“Ÿ, βˆ€ n, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ and f.coeff 0 βˆ‰ π“Ÿ ^ 2. In this file we gather miscellaneous results about Eisenstein polynomials.

Main definitions #

Main results #

Implementation details #

We also define a notion is_weakly_eisenstein_at requiring only that βˆ€ n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ. This makes certain results slightly more general and it is useful since it is sometimes better behaved (for example it is stable under polynomial.map).

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theorem polynomial.is_weakly_eisenstein_at_iff {R : Type u} [comm_semiring R] (f : polynomial R) (π“Ÿ : ideal R) :
f.is_weakly_eisenstein_at π“Ÿ ↔ βˆ€ {n : β„•}, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ
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structure polynomial.is_weakly_eisenstein_at {R : Type u} [comm_semiring R] (f : polynomial R) (π“Ÿ : ideal R) :
Prop

Given an ideal π“Ÿ of a commutative semiring R, we say that a polynomial f : R[X] is weakly Eisenstein at π“Ÿ if βˆ€ n, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ.

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structure polynomial.is_eisenstein_at {R : Type u} [comm_semiring R] (f : polynomial R) (π“Ÿ : ideal R) :
Prop

Given an ideal π“Ÿ of a commutative semiring R, we say that a polynomial f : R[X] is Eisenstein at π“Ÿ if f.leading_coeff βˆ‰ π“Ÿ, βˆ€ n, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ and f.coeff 0 βˆ‰ π“Ÿ ^ 2.

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theorem polynomial.is_eisenstein_at_iff {R : Type u} [comm_semiring R] (f : polynomial R) (π“Ÿ : ideal R) :
f.is_eisenstein_at π“Ÿ ↔ f.leading_coeff βˆ‰ π“Ÿ ∧ (βˆ€ {n : β„•}, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ) ∧ f.coeff 0 βˆ‰ π“Ÿ ^ 2
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theorem polynomial.is_weakly_eisenstein_at.map {R : Type u} [comm_semiring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_weakly_eisenstein_at π“Ÿ) {A : Type v} [comm_ring A] (Ο† : R β†’+* A) :
(polynomial.map Ο† f).is_weakly_eisenstein_at (ideal.map Ο† π“Ÿ)
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theorem polynomial.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree {R : Type u} [comm_ring R] {f : polynomial R} {S : Type v} [comm_ring S] [algebra R S] {p : R} {x : S} (hx : ⇑(polynomial.aeval x) f = 0) (hmo : f.monic) (hf : f.is_weakly_eisenstein_at (submodule.span R {p})) :
βˆƒ (y : S) (H : y ∈ algebra.adjoin R {x}), ⇑(algebra_map R S) p * y = x ^ (polynomial.map (algebra_map R S) f).nat_degree
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theorem polynomial.is_weakly_eisenstein_at.exists_mem_adjoin_mul_eq_pow_nat_degree_le {R : Type u} [comm_ring R] {f : polynomial R} {S : Type v} [comm_ring S] [algebra R S] {p : R} {x : S} (hx : ⇑(polynomial.aeval x) f = 0) (hmo : f.monic) (hf : f.is_weakly_eisenstein_at (submodule.span R {p})) (i : β„•) :
(polynomial.map (algebra_map R S) f).nat_degree ≀ i β†’ (βˆƒ (y : S) (H : y ∈ algebra.adjoin R {x}), ⇑(algebra_map R S) p * y = x ^ i)
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theorem polynomial.is_weakly_eisenstein_at.pow_nat_degree_le_of_root_of_monic_mem {R : Type u} [comm_ring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_weakly_eisenstein_at π“Ÿ) {x : R} (hroot : f.is_root x) (hmo : f.monic) (i : β„•) :
f.nat_degree ≀ i β†’ x ^ i ∈ π“Ÿ
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theorem polynomial.is_weakly_eisenstein_at.pow_nat_degree_le_of_aeval_zero_of_monic_mem_map {R : Type u} [comm_ring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_weakly_eisenstein_at π“Ÿ) {S : Type v} [comm_ring S] [algebra R S] {x : S} (hx : ⇑(polynomial.aeval x) f = 0) (hmo : f.monic) (i : β„•) :
(polynomial.map (algebra_map R S) f).nat_degree ≀ i β†’ x ^ i ∈ ideal.map (algebra_map R S) π“Ÿ
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theorem polynomial.scale_roots.is_weakly_eisenstein_at {R : Type u} [comm_ring R] (p : polynomial R) {x : R} {P : ideal R} (hP : x ∈ P) :
(p.scale_roots x).is_weakly_eisenstein_at P
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theorem polynomial.dvd_pow_nat_degree_of_evalβ‚‚_eq_zero {R : Type u} {A : Type u_1} [comm_ring R] [comm_ring A] {f : R β†’+* A} (hf : function.injective ⇑f) {p : polynomial R} (hp : p.monic) (x y : R) (z : A) (h : polynomial.evalβ‚‚ f z p = 0) (hz : ⇑f x * z = ⇑f y) :
x ∣ y ^ p.nat_degree
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theorem polynomial.dvd_pow_nat_degree_of_aeval_eq_zero {R : Type u} {A : Type u_1} [comm_ring R] [comm_ring A] [algebra R A] [nontrivial A] [no_zero_smul_divisors R A] {p : polynomial R} (hp : p.monic) (x y : R) (z : A) (h : ⇑(polynomial.aeval z) p = 0) (hz : z * ⇑(algebra_map R A) x = ⇑(algebra_map R A) y) :
x ∣ y ^ p.nat_degree
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theorem polynomial.monic.leading_coeff_not_mem {R : Type u} [comm_semiring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.monic) (h : π“Ÿ β‰  ⊀) :
f.leading_coeff βˆ‰ π“Ÿ
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theorem polynomial.monic.is_eisenstein_at_of_mem_of_not_mem {R : Type u} [comm_semiring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.monic) (h : π“Ÿ β‰  ⊀) (hmem : βˆ€ {n : β„•}, n < f.nat_degree β†’ f.coeff n ∈ π“Ÿ) (hnot_mem : f.coeff 0 βˆ‰ π“Ÿ ^ 2) :
f.is_eisenstein_at π“Ÿ
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theorem polynomial.is_eisenstein_at.is_weakly_eisenstein_at {R : Type u} [comm_semiring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_eisenstein_at π“Ÿ) :
f.is_weakly_eisenstein_at π“Ÿ
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theorem polynomial.is_eisenstein_at.coeff_mem {R : Type u} [comm_semiring R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_eisenstein_at π“Ÿ) {n : β„•} (hn : n β‰  f.nat_degree) :
f.coeff n ∈ π“Ÿ
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theorem polynomial.is_eisenstein_at.irreducible {R : Type u} [comm_ring R] [is_domain R] {π“Ÿ : ideal R} {f : polynomial R} (hf : f.is_eisenstein_at π“Ÿ) (hprime : π“Ÿ.is_prime) (hu : f.is_primitive) (hfd0 : 0 < f.nat_degree) :
irreducible f

If a primitive f satisfies f.is_eisenstein_at π“Ÿ, where π“Ÿ.is_prime, then f is irreducible.