Dummit and Foote Chapter 13, Exercise 2, page 519 reads as follows:

"Show that is irreducible over and let be a root.

Compute and in

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My attempt at this problem so far is as follows:

is irreducible over by Eisenstein's Criterion.

To compute I adopted the simple (but moderately ineffective) strategy of multiplying out and trying to use the fact that is a root of p(x) - that is to use the fact that .

Proceeding this way one finds the following:

Well, that does not seem to be going anywhere really! I must be missing something!

Can someone please help with the above and also help with the second part of the question ...

Peter