Let be an algebraic extension of , and let , where the are the conjugates of . Let , and denote by and the th conjugate of and (respectively). Show that if in , then in .
Suppose is an algebraic integer in . In particular, , so that for some polynomial .
Now is a root of some irreducible monic polynomial with rational integer coefficients. That is, . In particular, is a root of , so that the conjugates are also roots of . So for each , and thus is a root of a monic polynomial with rational integer coefficients. So is an algebraic integer in (in fact in ).