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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)2012-07-06
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    @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 $K$.2012-07-06

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