Let be a field, let be an extension of , and let be algebraic over . Recall that a polynomial is said to be minimal for over if , is monic, and has minimal degree among the nonzero polynomials having as a root.
Prove that minimal polynomials are unique.
Suppose and are minimal polynomials of over . By definition, and have the same degree and are both monic. Now is a root of , and has strictly smaller degree than do and . Multiplying by a unit if necessary, is monic. If , then we contradict the minimalness of (and ). Thus , and thus .