feat(number_theory/cyclotomic/primitive_roots): generalize norm_eq_one (

#12359)

We generalize `norm_eq_one`, that now computes the norm of any primitive `n`-th root of unity if `n ≠ 2`. We modify the assumption, that is still verified in the main case of interest `K = ℚ`.

From flt-regular

src/number_theory/cyclotomic/discriminant.lean

@@ -12,26 +12,23 @@ import ring_theory.discriminant
 We compute the discriminant of a `p`-th cyclotomic extension.
 
 ## Main results
-* `is_cyclotomic_extension.discr_odd_prime` : if `p` is an odd prime and
-  `is_cyclotomic_extension {p} K L`, then
-  `discr K (hζ.power_basis K).basis = (-1) ^ ((p - 1) / 2) * p ^ (p - 2)` for any primitive `n`-th
-  root `ζ`.
-
-## Implementation details
-We prove the result for any `K` such that `linear_ordered_field K` and
-`irreducible (cyclotomic p K)`. In practice these assumptions are satisfied for `K = ℚ`.
+* `is_cyclotomic_extension.discr_odd_prime` : if `p` is an odd prime such that
+  `is_cyclotomic_extension {p} K L` and `irreducible (cyclotomic p K)`, then
+  `discr K (hζ.power_basis K).basis = (-1) ^ ((p - 1) / 2) * p ^ (p - 2)` for any
+  `hζ : is_primitive_root ζ p`.
 
 -/
 
 universes u v
-variables {p : ℕ+} (k : ℕ) {K : Type u} {L : Type v} {ζ : L} [linear_ordered_field K] [field L]
+variables {p : ℕ+} (k : ℕ) {K : Type u} {L : Type v} {ζ : L} [field K] [field L]
 variables [algebra K L] [ne_zero ((p : ℕ) : K)]
 
 open algebra polynomial nat is_primitive_root
 
 namespace is_cyclotomic_extension
 
 local attribute [instance] is_cyclotomic_extension.finite_dimensional
+local attribute [instance] is_cyclotomic_extension.is_galois
 
 /-- If `p` is an odd prime and `is_cyclotomic_extension {p} K L`, then
 `discr K (hζ.power_basis K).basis = (-1) ^ ((p - 1) / 2) * p ^ (p - 2)`. -/
@@ -58,13 +55,11 @@ begin
     simp only [hζ.minpoly_eq_cyclotomic_of_irreducible hirr, aeval_add, _root_.map_mul, aeval_one,
       _root_.map_sub, aeval_X, minpoly.aeval, add_zero, aeval_nat_cast, aeval_X_pow] at H,
     replace H := congr_arg (algebra.norm K) H,
-    rw [monoid_hom.map_mul, hζ.sub_one_norm_prime hirr hodd, monoid_hom.map_mul,
-      monoid_hom.map_pow, norm_eq_one K hζ (odd_iff.2 (or_iff_not_imp_left.1
-      (nat.prime.eq_two_or_odd hp.out) hodd')), one_pow, mul_one, ← map_nat_cast
-      (algebra_map K L), norm_algebra_map, finrank _ hirr, totient_prime hp.out,
-      ← succ_pred_eq_of_pos hpos, pow_succ, mul_comm _ (p : K), coe_coe,
-      ← hζ.minpoly_eq_cyclotomic_of_irreducible hirr] at H,
-    simpa [(mul_right_inj' (cast_ne_zero.2 hp.out.ne_zero : (p : K) ≠ 0)).1 H],
+    rw [monoid_hom.map_mul, hζ.sub_one_norm_prime hirr hodd, monoid_hom.map_mul, monoid_hom.map_pow,
+      hζ.norm_eq_one hodd hirr, one_pow, mul_one, ← map_nat_cast (algebra_map K L),
+      norm_algebra_map, finrank _ hirr, totient_prime hp.out, ← succ_pred_eq_of_pos hpos, pow_succ,
+      mul_comm _ (p : K), coe_coe, ← hζ.minpoly_eq_cyclotomic_of_irreducible hirr] at H,
+    simpa [(mul_right_inj' $ ne_zero.ne ↑↑p).1 H],
     apply_instance },
   { apply_instance },
 end

src/number_theory/cyclotomic/primitive_roots.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Alex Best, Riccardo Brasca, Eric Rodriguez
 -/
 
+import data.pnat.prime
 import algebra.is_prime_pow
 import number_theory.cyclotomic.basic
 import ring_theory.adjoin.power_basis
@@ -31,15 +32,14 @@ in the implementation details section.
   `zeta n A B` is a primitive `n`-th root of unity.
 * `is_cyclotomic_extension.finrank`: if `irreducible (cyclotomic n K)` (in particular for
   `K = ℚ`), then the `finrank` of a cyclotomic extension is `n.totient`.
-* `is_primitive_root.norm_eq_one`: If `K` is linearly ordered (in particular for `K = ℚ`), the norm
-  of a primitive root is `1` if `n` is odd.
+* `is_primitive_root.norm_eq_one`: if `irreducible (cyclotomic n K)` (in particular for `K = ℚ`),
+  the norm of a primitive root is `1` if `n ≠ 2`.
 * `is_primitive_root.sub_one_norm_eq_eval_cyclotomic`: if `irreducible (cyclotomic n K)`
   (in particular for `K = ℚ`), then the norm of `ζ - 1` is `eval 1 (cyclotomic n ℤ)`, for a
   primitive root ζ. We also prove the analogous of this result for `zeta`.
-* `is_primitive_root.prime_ne_two_pow.sub_one_norm` : if `irreducible (cyclotomic (p ^ (k + 1)) K)`
+* `is_primitive_root.sub_one_norm_prime_ne_two` : if `irreducible (cyclotomic (p ^ (k + 1)) K)`
   (in particular for `K = ℚ`) and `p` is an odd prime, then the norm of `ζ - 1` is `p`. We also
   prove the analogous of this result for `zeta`.
-  gives a K-power basis for L given a primitive root `ζ`.
 * `is_primitive_root.embeddings_equiv_primitive_roots`: the equivalence between `L →ₐ[K] A`
   and `primitive_roots n A` given by the choice of `ζ`.
 
@@ -175,20 +175,52 @@ section norm
 namespace is_primitive_root
 
 variables [field L] {ζ : L} (hζ : is_primitive_root ζ n)
+variables {K} [field K] [algebra K L] [ne_zero ((n : ℕ) : K)]
+
+/-- This mathematically trivial result is complementary to `norm_eq_one` below. -/
+lemma norm_eq_neg_one_pow (hζ : is_primitive_root ζ 2) : norm K ζ = (-1) ^ finrank K L :=
+by rw [hζ.eq_neg_one_of_two_right , show -1 = algebra_map K L (-1), by simp, norm_algebra_map]
 
 include hζ
 
-/-- If `K` is linearly ordered (in particular for `K = ℚ`), the norm of a primitive root is `1`
+/-- If `irreducible (cyclotomic n K)` (in particular for `K = ℚ`), the norm of a primitive root is
+`1` if `n ≠ 2`. -/
+lemma norm_eq_one [is_cyclotomic_extension {n} K L] (hn : n ≠ 2)
+  (hirr : irreducible (cyclotomic n K)) : norm K ζ = 1 :=
+begin
+  by_cases h1 : n = 1,
+  { rw [h1, one_coe, one_right_iff] at hζ,
+    rw [hζ, show 1 = algebra_map K L 1, by simp, norm_algebra_map, one_pow] },
+  { replace h1 : 2 ≤ n,
+    { by_contra' h,
+      exact h1 (pnat.eq_one_of_lt_two h) },
+    rw [← hζ.power_basis_gen K, power_basis.norm_gen_eq_coeff_zero_minpoly, hζ.power_basis_gen K,
+      ← hζ.minpoly_eq_cyclotomic_of_irreducible hirr, cyclotomic_coeff_zero _ h1, mul_one,
+      hζ.power_basis_dim K, ← hζ.minpoly_eq_cyclotomic_of_irreducible hirr, nat_degree_cyclotomic],
+    exact neg_one_pow_of_even (totient_even (lt_of_le_of_ne h1 (λ h, hn (pnat.coe_inj.1 h.symm)))) }
+end
+
+/-- If `K` is linearly ordered, the norm of a primitive root is `1`
 if `n` is odd. -/
-lemma norm_eq_one [linear_ordered_field K] [algebra K L] (hodd : odd (n : ℕ)) : norm K ζ = 1 :=
+lemma norm_eq_one_of_linearly_ordered {K : Type*} [linear_ordered_field K] [algebra K L]
+  (hodd : odd (n : ℕ)) : norm K ζ = 1 :=
 begin
   haveI := ne_zero.of_no_zero_smul_divisors K L n,
   have hz := congr_arg (norm K) ((is_primitive_root.iff_def _ n).1 hζ).1,
   rw [←(algebra_map K L).map_one , norm_algebra_map, one_pow, map_pow, ←one_pow ↑n] at hz,
   exact strict_mono.injective hodd.strict_mono_pow hz
 end
 
-variables {K} [field K] [algebra K L] [ne_zero ((n : ℕ) : K)]
+lemma norm_of_cyclotomic_irreducible [is_cyclotomic_extension {n} K L]
+  (hirr : irreducible (cyclotomic n K)) : norm K ζ = ite (n = 2) (-1) 1 :=
+begin
+  split_ifs with hn,
+  { unfreezingI {subst hn},
+    convert norm_eq_neg_one_pow hζ,
+    erw [is_cyclotomic_extension.finrank _ hirr, totient_two, pow_one],
+    apply_instance },
+  { exact hζ.norm_eq_one hn hirr }
+end
 
 /-- If `irreducible (cyclotomic n K)` (in particular for `K = ℚ`), then the norm of
 `ζ - 1` is `eval 1 (cyclotomic n ℤ)`. -/
@@ -238,7 +270,7 @@ omit hζ
 
 /-- If `irreducible (cyclotomic (p ^ (k + 1)) K)` (in particular for `K = ℚ`) and `p` is an odd
 prime, then the norm of `ζ - 1` is `p`. -/
-lemma prime_ne_two_pow_sub_one_norm {p : ℕ+} [ne_zero ((p : ℕ) : K)] {k : ℕ}
+lemma sub_one_norm_prime_ne_two {p : ℕ+} [ne_zero ((p : ℕ) : K)] {k : ℕ}
   (hζ : is_primitive_root ζ ↑(p ^ (k + 1))) [hpri : fact (p : ℕ).prime]
   [is_cyclotomic_extension {p ^ (k + 1)} K L]
   (hirr : irreducible (cyclotomic (↑(p ^ (k + 1)) : ℕ) K)) (h : p ≠ 2) :
@@ -267,13 +299,13 @@ begin
   replace hζ : is_primitive_root ζ (↑(p ^ (0 + 1)) : ℕ) := by simp [hζ],
   haveI : ne_zero ((↑(p ^ (0 + 1)) : ℕ) : K) := ⟨by simp [ne_zero.ne ((p : ℕ) : K)]⟩,
   haveI : is_cyclotomic_extension {p ^ (0 + 1)} K L := by simp [hcyc],
-  simpa using prime_ne_two_pow_sub_one_norm hζ hirr h
+  simpa using sub_one_norm_prime_ne_two hζ hirr h
 end
 
 /-- If `irreducible (cyclotomic (2 ^ k) K)` (in particular for `K = ℚ`) and `k` is at least `2`,
 then the norm of `ζ - 1` is `2`. -/
-lemma sub_one_norm_pow_two [ne_zero (2 : K)] {k : ℕ} (hζ : is_primitive_root ζ (2 ^ k)) (hk : 2 ≤ k)
-  [is_cyclotomic_extension {2 ^ k} K L] (hirr : irreducible (cyclotomic (2 ^ k) K)) :
+lemma sub_one_norm_pow_two [ne_zero (2 : K)] {k : ℕ} (hζ : is_primitive_root ζ (2 ^ k))
+  (hk : 2 ≤ k) [is_cyclotomic_extension {2 ^ k} K L] (hirr : irreducible (cyclotomic (2 ^ k) K)) :
   norm K (ζ - 1) = 2 :=
 begin
   haveI : ne_zero (((2 ^ k : ℕ+) : ℕ) : K),
@@ -298,17 +330,17 @@ namespace is_cyclotomic_extension
 
 open is_primitive_root
 
-/-- If `K` is linearly ordered (in particular for `K = ℚ`), the norm of `zeta n K L` is `1`
+variables {K} (L) [field K] [field L] [algebra K L] [ne_zero ((n : ℕ) : K)]
+
+/-- If `irreducible (cyclotomic n K)` (in particular for `K = ℚ`), the norm of `zeta n K L` is `1`
 if `n` is odd. -/
-lemma norm_zeta_eq_one (K : Type u) [linear_ordered_field K] (L : Type v) [field L] [algebra K L]
-  [is_cyclotomic_extension {n} K L] (hodd : odd (n : ℕ)) : norm K (zeta n K L) = 1 :=
+lemma norm_zeta_eq_one [is_cyclotomic_extension {n} K L] (hn : n ≠ 2)
+  (hirr : irreducible (cyclotomic n K)) : norm K (zeta n K L) = 1 :=
 begin
   haveI := ne_zero.of_no_zero_smul_divisors K L n,
-  exact norm_eq_one K (zeta_primitive_root n K L) hodd,
+  exact norm_eq_one (zeta_primitive_root n K L) hn hirr,
 end
 
-variables {K} (L) [field K] [field L] [algebra K L] [ne_zero ((n : ℕ) : K)]
-
 /-- If `is_prime_pow (n : ℕ)`, `n ≠ 2` and `irreducible (cyclotomic n K)` (in particular for
 `K = ℚ`), then the norm of `zeta n K L - 1` is `(n : ℕ).min_fac`. -/
 lemma is_prime_pow_norm_zeta_sub_one (hn : is_prime_pow (n : ℕ))
@@ -333,7 +365,7 @@ begin
   { refine ⟨λ hzero, _⟩,
     rw [pow_coe] at hzero,
     simpa [ne_zero.ne ((p : ℕ) : L)] using hzero },
-  exact prime_ne_two_pow_sub_one_norm (zeta_primitive_root _ K L) hirr h,
+  exact sub_one_norm_prime_ne_two (zeta_primitive_root _ K L) hirr h,
 end
 
 /-- If `irreducible (cyclotomic p K)` (in particular for `K = ℚ`) and `p` is an odd prime,