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Auxiliary Polynomials in Number Theory by David Masser

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5

Irreducibility

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 Xx. If Y does not divide F0(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 ...

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