Inverse of Field Product with Inverse/Proof 1

Proof
By definition, a field is a non-trivial division ring whose ring product is commutative.

By definition, a division ring is a ring with unity such that every non-zero element is a unit.

Hence we can use Inverse of Division Product:
 * $\paren {\dfrac a b}^{-1} = \dfrac {1_R} {\paren {a / b}} = \dfrac b a$

which applies to the group of units of a comutative ring with unity.