Polynomial ring/Quotient field/Field/Exercise

Let ${}K[X]$ be the polynomial ring over a field ${}K$. Show that the set

${\left\{{\frac {P}{Q}}\mid P,Q\in K[X],\,Q\neq 0\right\}}$

with a suitable addition and multiplication is a field, where two fractions ${}{\frac {P}{Q}}$ and ${}{\frac {P'}{Q'}}$ are considered to be equal if ${}PQ'=P'Q$.

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