feat(data/polynomial/*): Specific root sets (#7510)

Adds lemmas for the root sets of a couple specific polynomials.

src/data/polynomial/field_division.lean

@@ -280,6 +280,24 @@ lemma mem_root_set [field k] [algebra R k] {x : k} (hp : p ≠ 0) :
   x ∈ p.root_set k ↔ aeval x p = 0 :=
 iff.trans multiset.mem_to_finset (mem_roots_map hp)
 
+lemma root_set_C_mul_X_pow {R S : Type*} [field R] [field S] [algebra R S]
+  {n : ℕ} (hn : n ≠ 0) {a : R} (ha : a ≠ 0) : (C a * X ^ n).root_set S = {0} :=
+begin
+  ext x,
+  rw [set.mem_singleton_iff, mem_root_set, aeval_mul, aeval_C, aeval_X_pow, mul_eq_zero],
+  { simp_rw [ring_hom.map_eq_zero, pow_eq_zero_iff (nat.pos_of_ne_zero hn), or_iff_right_iff_imp],
+    exact λ ha', (ha ha').elim },
+  { exact mul_ne_zero (mt C_eq_zero.mp ha) (pow_ne_zero n X_ne_zero) },
+end
+
+lemma root_set_monomial {R S : Type*} [field R] [field S] [algebra R S]
+  {n : ℕ} (hn : n ≠ 0) {a : R} (ha : a ≠ 0) : (monomial n a).root_set S = {0} :=
+by rw [←C_mul_X_pow_eq_monomial, root_set_C_mul_X_pow hn ha]
+
+lemma root_set_X_pow {R S : Type*} [field R] [field S] [algebra R S]
+  {n : ℕ} (hn : n ≠ 0) : (X ^ n : polynomial R).root_set S = {0} :=
+by { rw [←one_mul (X ^ n : polynomial R), ←C_1, root_set_C_mul_X_pow hn], exact one_ne_zero }
+
 lemma exists_root_of_degree_eq_one (h : degree p = 1) : ∃ x, is_root p x :=
 ⟨-(p.coeff 0 / p.coeff 1),
   have p.coeff 1 ≠ 0,

src/data/polynomial/ring_division.lean

@@ -534,6 +534,9 @@ rfl
   (0 : polynomial R).root_set S = ∅ :=
 by rw [root_set_def, polynomial.map_zero, roots_zero, to_finset_zero, finset.coe_empty]
 
+@[simp] lemma root_set_C [integral_domain S] [algebra R S] (a : R) : (C a).root_set S = ∅ :=
+by rw [root_set_def, map_C, roots_C, multiset.to_finset_zero, finset.coe_empty]
+
 instance root_set_fintype {R : Type*} [integral_domain R] (p : polynomial R) (S : Type*)
   [integral_domain S] [algebra R S] : fintype (p.root_set S) :=
 finset_coe.fintype _