Give an example of an extension $L : K$ such that $char(K) = p > 0$, $L : K$ is finite but not normal.
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Take $K=\mathbb{Z}_2(X)$ and $L=K(\sqrt[3]{X})$. Then the polynomial $f(T)=T^3-X\in K[T]$ is irreducible (because it has no roots in $K$), has one root in $L$, namely $\sqrt[3]{X}$, and I leave you the pleasure to prove that this is the only root of $f$ in $L$. |
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Here's a hint/general outline: We know that finite extensions of finite fields are normal, so we're going to need something a little more complicated. The simplest example here would be to take an extension of $\mathbb{F}_p(T)$, where $T$ is transcendental. Over the rationals, the canonical example is to adjoin an $n^{\text th}$ root ($n>2$) of some integer $a$, by adding a root of the equation $X^n-a=0$. In short, this will fail to be normal because $\mathbb{Q}$ does not include the $n^{\text th}$ roots of $1$. Try something similar with taking an $n^{\text th}$ root of $T$, where the $n^{\text th}$ roots of $1$ are not in $\mathbb{F}_p$. |
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