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**Sampras** **Theorem. **Suppose we have the conjugation isomorphism $\displaystyle \psi_{\alpha, \beta}: F(\alpha) \mapsto F(\beta) $ defined by $\displaystyle \psi_{\alpha, \beta}(a_0+a_{1} \alpha + \cdots + a_{n-1} \alpha^{n-1}) = a_0+a_{1} \beta + \cdots + a_{n-1} \beta^{n-1} $ (e.g. $\displaystyle \alpha $ and $\displaystyle \beta $ are conjugates). Then $\displaystyle \text{irr}(\alpha, F) = \text{irr}(\beta, F) $.

The notion is that we have to show that $\displaystyle \text{irr}(\alpha, F) $ divides $\displaystyle \text{irr}(\beta, F) $ **and** $\displaystyle \text{irr}(\beta, F) $ divides $\displaystyle \text{irr}(\alpha, F) $. But why can't we just stop and say $\displaystyle \text{irr}(\alpha, F) $ divides $\displaystyle \text{irr}(\beta, F) $? Because, by definition, if $\displaystyle \text{irr}(\alpha, F) $ divides $\displaystyle \text{irr}(\beta, F) $, wouldn't that imply that $\displaystyle \text{irr}(\alpha, F) = \text{irr}(\beta, F) $?