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Fields and homomorphisms

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Not sure why, but I am having trouble with this one. I'd really appreciate it if someone is willing to help here. Please see the attached file.

Thank you

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Please see the attachment.

(a) I define as , where and is a root of .
First, I claim that this mapping is well-defined.
Consider , then . Since , then . Then we have
So is well-defined.
Second, I claim that is a ...

Solution Summary

This provides an example of working with irreducible polynomials, fields, and homomorphisms.