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Consider $\mathbb{F}_3(\alpha)$ where $\alpha^3 - \alpha +1 = 0$ and $\mathbb{F}_3(\beta)$ where $\beta^3 - \beta^2 +1 =0$.

I know these two fields are isomorphic but I have difficulty buliding an isomorphism between them.

I know I have to determine where $\alpha$ is mapped to under the isomorphism map but I can't figure it out.

Any help is much appreciated.

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    Well, clearly $\alpha$ has to be mapped to something which is a root of $x^3-x+1=0$.2012-02-13
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    And there are only 27 elements of $\mathbb F_3(\beta)$ to check ...2012-02-13
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    Hi Chris, Suppose $\phi: \mathbb{F}_3(\alpha)\rightarrow\mathbb{F}_3(\beta)$ is the isomorphism. Then $\phi (\alpha)$ must be an expression in terms of $\beta$. Am I thinking about this the right way?!!?2012-02-13
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    I removed the "extraordinary isomorphisms" tag because for one, I feel like this is a rather ordinary isomorphism, and for two, I've never even heard that term and don't feel like it merits its own tag at this point. If someone feels this was hasty, please let me know.2012-02-13

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