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Let $K$ be any field and $s$ an indeterminate. 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 $t^n āˆ’ s^n$.

[Hint: first show that $s$ satisfies a poly of degree $n$ over $K(s^n)$; this gives $\leq$. Then show that $\{1, s, \ldots, s^{nāˆ’1}\}$ is Linearly Independent over $K(s^n)$; this gives $\geq$.]

I would really like some help with this please!

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    What's wrong with the hint you were given? Looks like it does pretty much all the work. – 2011-03-07

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