Counterexamples to Normal Extension

Let K\subseteq L\subseteq M be a tower of fields.

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

False. Let M be the algebraic closure of K=\mathbb{Q}. Let L=\mathbb{Q}(\sqrt[3]{2}).

Then M is certainly a normal extension of \mathbb{Q} since every irreducible polynomial in \mathbb{Q}[X] that has one root in M has all of its roots in M.

However consider X^3-2\in\mathbb{Q}[X]. It has one root (\sqrt[3]{2}) in L, but the other two complex roots are not in L. Thus L/K 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 L=\mathbb{Q}(\sqrt 2), K=\mathbb{Q}. Then L/K is normal since L is the splitting field of X^2-2 over \mathbb{Q}.

Let M=\mathbb{Q}(\sqrt 2,\sqrt[4]{2}). Then M/L is normal since M is the splitting field of X^2-\sqrt 2 over L.

However, M/K is not normal. The polynomial X^4-2 has a root in M (namely \pm\sqrt[4]{2}) but the other two complex roots are not in M.

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