Addition of Division Products in Field

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0$ and whose unity is $1$.

Let $a, b, c, d \in F$ such that $b \ne 0$ and $d \ne 0$.

Then:

$\dfrac a b + \dfrac c d = \dfrac {a d + b c} {b d}$

where $\dfrac x z$ is defined as $x \paren {z^{-1} }$.

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 Addition of Division Products:

$\dfrac a b + \dfrac c d = \dfrac {a \circ d + b \circ c} {b \circ d}$

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

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $3$: Field Theory: Definition and Examples of Field Structure: $\S 87 \delta$