Splitting field for X^3-5

Hi,

Let , the with real cube root of 5 and primitive root of 1.

because is irreducible.

But what is ?

because and yet is a subfield of . Also the degree is because has as a root. So the possibilities are 2 or 3.

If it's 2. Then because the min poly has to be monic and

I think that we have to express as a linear combination (with rational coefficients) of and .

I can't see how do to this?

Thanks for any help!

Re: Splitting field for X^3-5

The roots of are whereby and have minimal polynomial

Re: Splitting field for X^3-5

Of course, I made a stupid mistake:

!

thanks