I am having trouble with this problem. It would be great if someone can help.
I want to prove that where is the cyclotomic field of 5th roots of unity, and
first see that is a root of the polynomial which is irreducible over by Eisenstein criterion. thus since we'll be done if we prove that
now suppose then from we'll get: which after simplifying
gives us: thus: therefore is a root of we actually proved that the roots of are exactly
thus there exists such that
Thank you very much for your help, NonCommAlg. So I understand that you let . Then squaring this equation twice and manipulate to get the polynomial. It is also the minimal polynomial correct? But I don't understand the notation you use below. Does it mean degree of
since