exist and an integer such that but then for some which is obviously nonsense.
so the integral closure of in is not in fact is integrally closed in i.e. the integral closure of in is itself: clearly for every there exist and such that
if is integral over then is also integral over so there exist and an integer such that
that would give us for some which is possible only if because, since there is no in therefore
which gives us thus and the proof is complete.