unfold field_inverse in |- *.
auto with algebra.
Qed.
Hint Resolve FIELD_comp FIELD_inverse_r FIELD_inverse_l FIELD_unit_non_zero:
algebra.

Lemma FIELD_unit_inverse : Equal (field_inverse (ring_unit K)) (ring_unit K).
apply Trans with (ring_mult (field_inverse (ring_unit K)) (ring_unit K));
auto with algebra.