Polynomial ring/Quotient field/Field/Exercise

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

$\{\frac{P}{Q}\mid P,Q\in K[X],Q\ne 0\}$

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

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