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Abstract Algebra : Fields, Rings and Domains

I) Show that if D is an integral domain then D[x] is never a field.

ii) Is the assumption "D is an integral domain" needed here? That is, does the conclusion hold if D is merely assumed to be a ring?

Solution Preview

1) Proof:
Suppose D[x] is a field. We consider f(x)=x in D[x]. then f(x) has an inverse g(x) in D[x]. Suppose g(x)=a_n*x^n+...+a_1*x+a_0.
Since f(x)*g(x)=1, then we have ...

Solution Summary

Fields and rings are investigated. The response received a rating of "5" from the student who originally posted the question.