Thread: Field extension

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

Originally Posted by dangkhoa

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 .