# Math Help - Field extension

1. ## 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

2. 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 $F=K(s^n)$. Then, $\{1_{F}, s, s^2, \cdots, s^{n-1}\}$ is a basis of vector space $K(s)$ over $F=K(s^n)$, which implies that every element of $K(s)$ can be written uniquely in the form of $c_0 + c_1s \cdots c_{n-1}s^{n-1} (c_i \in F)$. Thus $[K(s):K(s^n)] = n$.

$f(t)=t^n-s^n \in K(s^n)[t]$ is an irreducible monic polynomial of degree n over $K(s^n)$ such that $f(s)=0$.