Example of an non-normal inseparable field extension I was asked to prove or disprove that if a field extension is not normal then it is separable. I can't see why this would be true so I want to disprove it with an example of a non-normal inseparable field extension. I am wondering if there is a simple example of such an extension, or even any examples at all. I know that any inseparable extension $L/K$ has to be such that $K$ is infinite and of positive characteristic, so a good place to start would be $K=\mathbb F_p(T)$, but I'm unsure how to prove that an extension of this is inseparable, and even unsure how to come up with a non-normal example, since my usual approach is to exploit that $K$ is a subfield of $\mathbb R$ (so e.g. $\mathbb Q(\sqrt[3]2)/\mathbb Q$ is not normal as $x^3-2$ has $\sqrt[3]2$ as a root but the other two roots are in $\mathbb C \setminus \mathbb R$ and $\mathbb Q(\sqrt[3]2)\subset \mathbb R$). However with $K$ as above, $K \nsubseteq \mathbb R. $
 A: Added
All extensions are separable in characteristic $0$, and all finite fields are separable extensions of $\mathbb{F}_p$.
end addition
A standard example of non-separable extension is $L:=\frac{\mathbb{F}_5(t)[X]}{(X^5-t)}=\mathbb{F}_5(\sqrt[5]{t})$ over $\mathbb{F}_5(t)$.
Now, take $K:=\frac{L[X]}{X^3-\sqrt[5]{t}}=\mathbb{F}_5(\sqrt[15]{t})$. It keeps being inseparable over $\mathbb{F}_5(t)$, but it's not normal, because if it contained all three distinct roots of $X^3-\sqrt[5]{t}$, then it would contain also all distinct roots of $X^3-1$, hence the whole $\mathbb{F}_{25}$.
But every element of $K$ is written as $\frac{p(\sqrt[15]{t})}{q(\sqrt[15]{t})}$, $\ p,q\in\mathbb{F}_5[X]$ coprime polynomials. Hence, there would exist coprime polynomials $p,q\in\mathbb{F}_5[X]$ satisfying $\left(\frac{p(\sqrt[15]{t})}{q(\sqrt[15]{t})}\right)^2+\left(\frac{p(\sqrt[15]{t})}{q(\sqrt[15]{t})}\right)+1=0$.
But since $\sqrt[15]{t}$ is, by definition, transcendent over $\mathbb{F}_5$, the evaluation $X\mapsto\sqrt[15]{t}$ is injective. But no rational function $f\in\mathbb{F}_5(X)$ satisfies $f^2+f+1$.
