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

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We prove the statement by induction over . For the statement holds. So suppose that and that the statement is already proven for smaller degrees. Let be a zero of (if does not have a zero at all, we are done anyway). Hence, by fact and the degree of is , so we can apply to the induction hypothesis. The polynomial has at most zeroes. For we have . This can be zero, due to fact, only if one factor is , so the zeroes of are or a zero of . Hence, there are at most zeroes of .