Polynomials f(x) & g(x) in Z[x] Relatively Prime in Q[x]

esisk · Dec 10, 2009

trying to show that polynomials f(x), g(x) in Z[x] are relatively prime in Q[x] iff the ideal they generate in Z[x] contains an integer.Thanks .Not homework

rochfor1 · Dec 10, 2009

Z[x] contains an integer => f(x), g(x) coprime in Q[z] is easy. If f(x) and g(x) are coprime in Q[x], then there exist a(x) and b(x) in Q[x]. such that a(x) f(x) + b(x) g(x) = 1. There is an integer n such that n a(x) and n b(x) have coefficients in Z (think about why this is). Then n a(x) f(x) + n b(x) g(x) = n, so the ideal generated by f and g in Z[x] contains an integer.

esisk · Dec 10, 2009

This is quite Rochfor1, thank you. And, yes, I will be able to do the other implication.

esisk · Dec 10, 2009

Sorry, I meant "quite clear". Thanks

Polynomials f(x) & g(x) in Z[x] Relatively Prime in Q[x]

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