feat(algebra/field/basic): div_neg_self, neg_div_self (#11438)

I think these two lemmas are useful as `simp` lemmas, but they don't
seem to be there already.

src/algebra/field/basic.lean

@@ -97,6 +97,12 @@ by simp [neg_div]
 lemma neg_div_neg_eq (a b : K) : (-a) / (-b) = a / b :=
 by rw [div_neg_eq_neg_div, neg_div, neg_neg]
 
+@[simp] lemma div_neg_self {a : K} (h : a ≠ 0) : a / -a = -1 :=
+by rw [div_neg_eq_neg_div, div_self h]
+
+@[simp] lemma neg_div_self {a : K} (h : a ≠ 0) : (-a) / a = -1 :=
+by rw [neg_div, div_self h]
+
 @[field_simps] lemma div_add_div_same (a b c : K) : a / c + b / c = (a + b) / c :=
 by simpa only [div_eq_mul_inv] using (right_distrib a b (c⁻¹)).symm