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

• esisk
In summary, polynomials f(x) and g(x) in Z[x] are relatively prime in Q[x] if and only if the ideal they generate in Z[x] contains an integer. This is shown by the fact that 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. By choosing an integer n such that n a(x) and n b(x) have coefficients in Z, it follows that n a(x) f(x) + n b(x) g(x) = n, showing that the ideal generated by f and g in Z[x
esisk
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

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.

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

Sorry, I meant "quite clear". Thanks

## 1. What does it mean for two polynomials to be relatively prime?

Two polynomials, f(x) and g(x), are relatively prime if the only common factor they share is a constant. In other words, there is no polynomial that can divide both f(x) and g(x) without leaving a remainder.

## 2. How do I determine if two polynomials are relatively prime in Z[x]?

In order to determine if two polynomials, f(x) and g(x), are relatively prime in Z[x], you can use the Euclidean algorithm. This algorithm involves dividing the larger polynomial by the smaller one and continuing the process until the remainder is equal to 0. If the final remainder is 1, then the polynomials are relatively prime in Z[x].

## 3. Can two polynomials be relatively prime in Q[x] but not in Z[x]?

Yes, two polynomials can be relatively prime in Q[x] but not in Z[x]. This is because the coefficients in Q[x] can be rational numbers, whereas the coefficients in Z[x] must be integers. Therefore, the Euclidean algorithm may yield a different result when applied to the same polynomials in these two different polynomial rings.

## 4. What are the implications of two polynomials being relatively prime in Q[x]?

If two polynomials, f(x) and g(x), are relatively prime in Q[x], it means that they do not have any common factors other than a constant. This can be useful in simplifying expressions or solving equations involving these polynomials. It also means that the greatest common divisor of these polynomials is equal to 1.

## 5. Can two polynomials that are relatively prime in Q[x] also be relatively prime in R[x]?

Yes, two polynomials can be relatively prime in Q[x] and also in R[x]. This is because the coefficients in R[x] can be real numbers, which includes rational numbers. Therefore, the Euclidean algorithm may yield the same result when applied to the same polynomials in these two different polynomial rings.

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