A sufficient condition for the conjugates over a field of an algebraic element to be distinct

Let F be a field and \alpha be algebraic of degree n over F. Suppose there exists an element \lambda \in F(\alpha) such that \lambda \alpha = \beta \in F. Show that the conjugates of \alpha over F(\alpha) are distinct.


Certainly we have F(\lambda) \subseteq F(\alpha). Now \lambda^{-1} \in F(\lambda), so \alpha = \beta\lambda^{-1} \in F(\lambda). (Recall that \beta \in F, so that \beta\lambda^{-1} can be written uniquely as a polynomial in \lambda.) So \alpha \in F(\lambda), and we have F(\alpha) \subseteq F(\lambda). Hence F(\alpha) = F(\lambda).

By Theorem 5.10 in TAN, because F(\alpha) = F(\lambda), the conjugates of \lambda over F(\alpha) are distinct.

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