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 $M$ .

Author: mathtuition88

Math and Education Blog View all posts by mathtuition88

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