Let be a tower of fields.

**Q1) If M/K is a normal extension, is L/K a normal extension?**

False. Let be the algebraic closure of . Let .

Then is certainly a normal extension of since every irreducible polynomial in that has one root in has all of its roots in .

However consider . It has one root in , but the other two complex roots are not in . Thus is not a normal extension.

**Q2) If M/L and L/K are both normal extensions, is M/K a normal extension? (i.e. is normal extension transitive?)**

False. Let , . Then is normal since is the splitting field of over .

Let . Then is normal since is the splitting field of over .

However, is not normal. The polynomial has a root in (namely ) but the other two complex roots are not in .