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Math Help - Field Extensions

  1. #1
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    Field Extensions

    1. Let K be a field extension of F of prime degree p and let \alpha be an element of K which is not in F. Prove that K = F( \alpha)

    im not sure how to solve this
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  2. #2
    Senior Member JaneBennet's Avatar
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    We have p=[K:F]=[K:F(\alpha)]\times[F(\alpha):F]. Thus [F(\alpha):F] = p or 1. Since F(\alpha)\supsetneq F, we can’t have [F(\alpha):F] = 1; therefore [F(\alpha):F] = p \Rightarrow [K:F(\alpha)] = 1, i.e. K=F(\alpha).
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