Theorem 4.7 of the previous chapter can be reformulated in terms of reducibility. It implies that there are at most finitely many *y* in Z for which there exists *x* in Z with *F*(*x*, *y*) = 0; that is, *F*(*X*, *y*) has the factor *X* − *x*. If *Y* does not divide *F*_{0}(*X*, *Y*), then it can be shown fairly easily using the concept of integrality (see Exercise 5.2) that this is almost equivalent to *F*(*X*, *y*) having a linear factor over Q. That may remind us of the Hilbert Irreducibility Theorem for unrestricted F(*X*, *Y*) in Q[*X*, *Y*] irreducible over Q; namely that there exist infinitely many *y* in Q such that F(*X*, *y*) is irreducible over Q. By pushing the techniques of the previous chapter a little further, we can prove the following stronger version of this ...

Start Free Trial

No credit card required