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Spacepotato (talk | contribs) →Proof: clarify |
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== Proof ==
Suppose first that <math>F</math> is infinite. If <math>F(\alpha, \beta) \supsetneq F(\alpha + c \beta)</math> for some <math>c\in F</math>, the latter field must not contain <math>\beta</math> (otherwise <math>\alpha = (\alpha + c \beta) - c\beta</math> would also be in it). Therefore, we may extend the inclusion <math>F(\alpha + c \beta) \to F(\alpha, \beta)</math> to an <math>F</math>-automorphism <math>\sigma</math> that sends <math>\beta</math> to a different root of the minimal polynomial of <math>\beta</math> over <math>F(\alpha + c \beta)</math>, since they are all different by separability (the polynomial in question is a divisor of the minimal polynomial of <math>\beta</math> over <math>f</math>). (Possibly we have to enlarge <math>F(\alpha, \beta)</math>.) We then have
:<math>\alpha + c \beta = \sigma(\alpha + c \beta) = \sigma(\alpha) + c \sigma(\beta)</math> and therefore <math>c = \frac{\sigma(\alpha) - \alpha}{\beta - \sigma(\beta)}</math>.
Since there are only finitely many possibilities for <math>\sigma</math>, there are only finitely many possibilities for the value of <math>c</math>. For all other values of <math>c\in F</math> then <math>F(\alpha,\beta) = F(\alpha+c\beta)</math>.
For the case where <math>F</math> is finite, we simply use a primitive root, which then generates the field extension.
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