A fact about divisibility in a splitting field

Let E = \mathbb{Q}(\theta) be an algebraic extension of \mathbb{Q}, and let K = \mathbb{Q}(\theta_1,\ldots,\theta_k), where the \theta_i are the conjugates of \theta. Let \alpha,\beta \in E, and denote by \alpha_i and \beta_i the ith conjugate of \alpha and \beta (respectively). Show that if \beta|\alpha in E, then \beta_i|\alpha_i in E.


Suppose \alpha/\beta is an algebraic integer in K. In particular, \alpha/\beta \in F(\theta), so that \alpha/\beta = r(\theta) for some polynomial r(x).

Now \alpha/\beta is a root of some irreducible monic polynomial h(x) with rational integer coefficients. That is, h(r(\theta)) = 0. In particular, \theta is a root of h \circ r, so that the conjugates \theta_i are also roots of h \circ r. So h(r(\theta_i)) = 0 for each \theta_i, and thus \alpha_i/\beta_i = r(\theta_i) is a root of a monic polynomial with rational integer coefficients. So \alpha_i/\beta_i is an algebraic integer in K (in fact in \mathbb{Q}(\theta_i)).

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