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esisk
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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
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.
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].
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.
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.
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.