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Fields and Ideals

Let F be a field and f(x), g(x) be elements of F[x]. Show that f(x) divides g(x) if and only if g(x) is an element of < f(x) >.

Note that < f(x) > is an ideal.

Below is a problem from an undergraduate course in Abstract Algebra. The book we use is titled "A First Course in Abstract Algebra" by John B. Fraleigh. We have just started Ring and Field Theory. If you are able to solve the following problem, please detail any theorems or definitions in your answers.

Solution Preview

I'll use shorthand to denote f(x) by f, etc.

If f divides g , then g = ...

Solution Summary

Fields and ideals are investigated.

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