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Polynomial rings

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Please see the attachment.

(a) For , we can rewrite it as .
If is reducible, then for some and in . I claim that . If , then we have for some . Then is a zero point of . Thus for any . But we note that ...

Solution Summary

This shows how to prove that given polynomials are irreducible.

See Also This Related BrainMass Solution

Ring Theory: Polynomial Rings

Consider the polynomial ring R=Q[x].

(a) show that I = {f(x) (x^3-6x+7)+g(x) (x+4) | f(x), g(x) in R} is an ideal of R.

(b) We have seen that R is a principle ideal domain. That is, every ideal is generated by a single element of R. Find h(x) in R so that I = {f(x)h(x) | f(x) in R}.

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