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feat(data/polynomial/unit_trinomial): Irreducibility of X^n-X-1 (#15318)
This PR adds a proves irreducibility of X^n-X-1, superseding #6421.
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/- | ||
Copyright (c) 2022 Thomas Browning. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Thomas Browning | ||
-/ | ||
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import data.polynomial.unit_trinomial | ||
import tactic.linear_combination | ||
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/-! | ||
# Irreducibility of Selmer Polynomials | ||
This file proves irreducibility of the Selmer polynomials `X ^ n - X - 1`. | ||
## Main results | ||
- `polynomial.selmer_irreducible`: The Selmer polynomials `X ^ n - X - 1` are irreducible. | ||
TODO: Show that the Selmer polynomials have full Galois group. | ||
-/ | ||
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namespace polynomial | ||
open_locale polynomial | ||
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variables {n : ℕ} | ||
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lemma X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬ (z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0) := | ||
begin | ||
rintros ⟨h1, h2⟩, | ||
replace h3 : z ^ 3 = 1, | ||
{ linear_combination (1 - z - z ^ 2 - z ^ n) * h1 + (z ^ n - 2) * h2 }, -- thanks polyrith! | ||
have key : z ^ n = 1 ∨ z ^ n = z ∨ z ^ n = z ^ 2, | ||
{ rw [←nat.mod_add_div n 3, pow_add, pow_mul, h3, one_pow, mul_one], | ||
have : n % 3 < 3 := nat.mod_lt n zero_lt_three, | ||
interval_cases n % 3; simp only [h, pow_zero, pow_one, eq_self_iff_true, or_true, true_or] }, | ||
have z_ne_zero : z ≠ 0 := | ||
λ h, zero_ne_one ((zero_pow zero_lt_three).symm.trans (show (0 : ℂ) ^ 3 = 1, from h ▸ h3)), | ||
rcases key with key | key | key, | ||
{ exact z_ne_zero (by rwa [key, self_eq_add_left] at h1) }, | ||
{ exact one_ne_zero (by rwa [key, self_eq_add_right] at h1) }, | ||
{ exact z_ne_zero (pow_eq_zero (by rwa [key, add_self_eq_zero] at h2)) }, | ||
end | ||
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lemma X_pow_sub_X_sub_one_irreducible (hn1 : n ≠ 1) : irreducible (X ^ n - X - 1 : ℤ[X]) := | ||
begin | ||
by_cases hn0 : n = 0, | ||
{ rw [hn0, pow_zero, sub_sub, add_comm, ←sub_sub, sub_self, zero_sub], | ||
exact associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X }, | ||
have hn : 1 < n := nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn0, hn1⟩, | ||
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := | ||
by simp only [trinomial, C_neg, C_1]; ring, | ||
rw hp, | ||
apply is_unit_trinomial.irreducible_of_coprime' ⟨0, 1, n, zero_lt_one, hn, -1, -1, 1, rfl⟩, | ||
rintros z ⟨h1, h2⟩, | ||
apply X_pow_sub_X_sub_one_irreducible_aux z, | ||
rw [trinomial_mirror zero_lt_one hn (-1 : ℤˣ).ne_zero (1 : ℤˣ).ne_zero] at h2, | ||
simp_rw [trinomial, aeval_add, aeval_mul, aeval_X_pow, aeval_C] at h1 h2, | ||
simp_rw [units.coe_neg, units.coe_one, map_neg, map_one] at h1 h2, | ||
replace h1 : z ^ n = z + 1 := by linear_combination h1, | ||
replace h2 := mul_eq_zero_of_left h2 z, | ||
rw [add_mul, add_mul, add_zero, mul_assoc (-1 : ℂ), ←pow_succ', nat.sub_add_cancel hn.le] at h2, | ||
rw h1 at h2 ⊢, | ||
exact ⟨rfl, by linear_combination -h2⟩, | ||
end | ||
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lemma X_pow_sub_X_sub_one_irreducible_rat (hn1 : n ≠ 1) : irreducible (X ^ n - X - 1 : ℚ[X]) := | ||
begin | ||
by_cases hn0 : n = 0, | ||
{ rw [hn0, pow_zero, sub_sub, add_comm, ←sub_sub, sub_self, zero_sub], | ||
exact associated.irreducible ⟨-1, mul_neg_one X⟩ irreducible_X }, | ||
have hp : (X ^ n - X - 1 : ℤ[X]) = trinomial 0 1 n (-1) (-1) 1 := | ||
by simp only [trinomial, C_neg, C_1]; ring, | ||
have hn : 1 < n := nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn0, hn1⟩, | ||
have h := (is_primitive.int.irreducible_iff_irreducible_map_cast _).mp | ||
(X_pow_sub_X_sub_one_irreducible hn1), | ||
{ rwa [polynomial.map_sub, polynomial.map_sub, polynomial.map_pow, polynomial.map_one, | ||
polynomial.map_X] at h }, | ||
{ exact hp.symm ▸ (trinomial_monic zero_lt_one hn).is_primitive }, | ||
end | ||
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end polynomial |