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What field axioms and then properties can I use to prove: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ for any elements $a, b, c, d$ in a field, $b, d \neq 0$. I am trying to think more abstractly, however the only property that comes to my mind is that $\frac{a}{b}$ is just the solution to the equation $a = bx$ and $b = bx$ to use the multiplicative identity.

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You can use "inverse" notation. We are interested in $ab^{-1}+cd^{-1}$. This is $(ab^{-1}+cd^{-1})(bd)(bd)^{-1}$. But a straightforward calculation shows that $(ab^{-1}+cd^{-1})(bd)=ad+cb$, and we are finished.

If the definition of $\frac{a}{b}$ that you are using is that it is the solution of the equation $bx=a$, then as a preliminary step show that $x=ab^{-1}$.

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$a=bx$, $c=dy$, $ad+bc=(bx)d+b(dy)=bd(x+y)$ seems to do what you want.

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Hint $\ $ In a field, if $\rm\,u\ne 0\,$ then $\rm\,u\,x = v\,$ has unique solution $\rm\,x = u^{-1} v.\,$ Hence, by uniqueness, we deduce that $\rm\:a/b + c/d\, = \,(ad+bc)/bd\:$ since both are solutions of $\rm\: bd\, x \,=\, ad+bc.$

Remark $\ $ Quite frequently, uniqueness theorems provide powerful tools for proving equalities.