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$X^4 -4$ has a root in $\Bbb Q(2^{1/2})$ but does not split in $\Bbb Q(2^{1/2})$ implying that $\Bbb Q(2^{1/2})$ is not a normal extension of $\Bbb Q$ according to most definitions. But $\Bbb Q(2^{1/2})$ is considered a normal extension of $\Bbb Q$ by everybody. What am I missing here?

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Please don't yell. (All caps is considered yelling) – Arturo Magidin Jul 6 '12 at 22:31
@DonL.: An algebraic field extension $K$ of $F$ is normal if and only if whenever an irreducible polynomial $f(x)\in F[x]$ has at least one root in $K$, it splits in $K$. – Arturo Magidin Jul 6 '12 at 22:34

You are missing the fact that $x^4-4$ is not irreducible over $\mathbb{Q}$: $x^4-4 = (x^2-2)(x^2+2)$.

The definition you have in mind says that if $K/F$ is algebraic, then $K$ is normal if and only if every irreducible $f(x)\in F[x]$ that has at least one root in $K$ actually splits in $K$. Your test polynomial is not irreducible.

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Thanks. That clears up a lot of things for me. John Heaney – John Heaney Jul 6 '12 at 22:51

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