Polynomial ring/Field/Zeroes/Number/Fact/Proof

Proof

We prove the statement by induction over ${}d$. For ${}d=0,1$ the statement holds. So suppose that ${}d\geq 2$ and that the statement is already proven for smaller degrees. Let ${}a$ be a zero of ${}P$ (if ${}P$ does not have a zero at all, we are done anyway). Hence, ${}P=Q(X-a)$ by fact and the degree of ${}Q$ is ${}d-1$, so we can apply to ${}Q$ the induction hypothesis. The polynomial ${}Q$ has at most ${}d-1$ zeroes. For ${}b\in K$ we have ${}P(b)=Q(b)(b-a)$. This can be zero, due to fact  (5), only if one factor is ${}0$, so the zeroes of ${}P$ are ${}a$ or a zero of ${}Q$. Hence, there are at most ${}d$ zeroes of ${}P$.

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