A fact about simple algebraic field extensions

Let F be a field and let E be an extension of F. Let \alpha,\beta \in E be algebraic over F with minimal polynomials a(x) and b(x), respectively. Show that (F(\alpha))(\beta) = (F(\beta))(\alpha).


Recall that F(\alpha) has a kind of universal property with respect to fields containing an element whose minimal polynomial over F is a(x). Since (F(\beta))(\alpha) contains an element which is algebraic over F with minimal polynomial a(x), we have an injective ring homomorphism F(\alpha) \rightarrow (F(\beta))(\alpha) fixing F and \alpha. Similarly, we have an injective map (F(\alpha))(\beta) \rightarrow (F(\beta))(\alpha) fixing F, \alpha, and \beta. Since every element of (F(\beta))(\alpha) is a linear combination of \alpha^i\beta^j, this map is the identity. So (F(\alpha))(\beta) = (F(\beta))(\alpha).

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