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I need to show

$\mathbb{F}_2(x)$ is not separable in the field $\mathbb{F}_2(x^2)$. Here is what I know so far : char($\mathbb{F_2}$)=2 and we have that it is a perfect field. I also know that any irreducible polynomial on a perfect field is separable.

Can anyone help?

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    so you don't know the definition of a separable extension ? – 2017-02-10
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    @mercio seperable if all its roots are simple. – 2017-02-10
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    What is the minimal polynomial of $x$ over $\mathbb{F}_2(x^2)$ ? Is this polynomial separable (what is its formal derivative) ? – 2017-02-10
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    Is the minimal polynomial of $x$ over $\mathbb{F}_2(x^2) = x^2?$ Thus we have $(2x, x^2)=x$, no it is not separable? – 2017-02-10
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    Not quite. Note that if you want minimal polynomials of algebraic elements over $\Bbb F_2(x^2)$, then $x$ _cannot_ be the variable you use. You have to use another, like $t$. So the minimal polynomial is, for instance, $t^2-x^2$. – 2017-02-10
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    so $(t^2-x^2, 2t)=t$. Then it only shows that $f(t)=t^2-x^2$ is not separable in $\mathbb{F}_2(x)$. Then what? – 2017-02-10
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    @Arthur what should I do – 2017-02-11
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    What is the derivative of $f$ (with respect to $t$)? What is the definition of "separable"? Do those fit together? Alternatively, over the algebraic closure of $\Bbb F_2(x^2)$, we have $f(t) = (x+t)(x-t) = (x+t)^2$, which clearly has multiple roots. – 2017-02-11
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    Things will be much clearer if you set $x^2=t$ and show that $\Bbb F_2(\sqrt{t\,}\,)$ is inseparable over $\Bbb F_2(t)$. – 2017-02-12

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