Let . Then, is a basis of vector space over , which implies that every element of can be written uniquely in the form of . Thus .
is an irreducible monic polynomial of degree n over such that .
Let K(s) be the field of rational functions in one variable s, i.e the field of fractions of K[s]. Then K(s) is a field extension of K(s^n).
Prove that [K(s):K(s^n)] = n. Hence show that the minimum polynomial of s over K(s^n) is (t^n) - (s^n)
How do you prove it?
Thank you very much
Let . Then, is a basis of vector space over , which implies that every element of can be written uniquely in the form of . Thus .
is an irreducible monic polynomial of degree n over such that .