2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76378We prove that any Galois extension of commutative rings with normal basis and abelian Galois group of odd order has a self dual normal basis. Also we show that if S/R is an unramified extension of number rings with Galois group of odd order and $R$ is totally real then the normal basis does not exist for S/R.Number Theorytext