Find a primitive generator for over .

I found the polynomial for the root , which is . But I'm having trouble showing the polynomial is irreducible. Some help please.

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- Apr 21st 2009, 06:04 PMdori1123irreducibility
Find a primitive generator for over .

I found the polynomial for the root , which is . But I'm having trouble showing the polynomial is irreducible. Some help please. - Apr 21st 2009, 08:42 PMGammaHere is an idea
I am not sure how to show immediately that it is irreducible; however here is another idea.

We know that:

But it is easy to show each of the intermediary steps is a degree 2 extension with irreducible minimal polynomials all irreducible by Eisenstein with p = 5, 3, 2 respectively. So that means the total extension is:

Since you have found a monic polynomial of degree 8 that has this as a root it must be irreducible. Kind of roundabout, but hope it helps.