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Let the $ \mathbb F_{p}$ denote the finite field of $\mathbb Z/ p \mathbb Z$ and $\overline{ \mathbb F_{p}}$ its algebraic closure.

Now let $f(x)=X^p- b \in \overline{ \mathbb F_{p}}[x]$. I want to $f(x)$ has exactly one root. I know all is derivative are zeroes. How does that prove it has exactly one root?

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    Well, the first derivative is $pX^{p-1}$, which is the zero polynomial.2012-04-01

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Hint: There is an $\alpha\in \overline{\mathbb{F}_p}$, such that $f(\alpha)=0$. Show $f(X)=(X-\alpha)^p$.

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Working with derivatives is not a good idea here because the characteristic of the fields is positive.

As a hint: Do you know about freshman's dream?

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    Yes, I can show that.Then I can prove $X^p-b=(X-a)^p$ where a is one solution of the equation $X^p-b$.2012-04-01