/sci/ - Science & Math

Am I being retarded or how is this question not wrong - [math]L/L_0[/math] being Galois implies that it's normal, ie that any polynomial over [math]L_0[/math] that has a root in [math]L[/math] splits over [math]L[/math]. But any polynomial over [math]F[/math] is a polynomial over [math]L_0[/math] so it splits over [math]L[/math], hence actually [math]L/F[/math] is a priori normal - but that is not an assumption.