Inseparable extension of rational function field

I am to construct an extension E of k:=F(x), where F=Z/2, such that E/k is not separable. I saw in an article by Keith Conrad that t^2-x in k[t] is not separable, because if a is a root, then a^2=x so that t^2 -x = t^2 - a^2 = (t-a)^2 (remember we're in characteristic 2). Can I use this somehow to construct the desired extension? I mean, k[t]/k itself is not an extension, a field extension that is, which I can only assume is what is meant. I was thinking k(t) at first, but I really don't know. I would really appreciate help with this one, thanks.

Re: Inseparable extension of rational function field

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**spudwish** I am to construct an extension E of k:=F(x), where F=Z/2, such that E/k is not separable. I saw in an article by Keith Conrad that t^2-x in k[t] is not separable, because if a is a root, then a^2=x so that t^2 -x = t^2 - a^2 = (t-a)^2 (remember we're in characteristic 2). Can I use this somehow to construct the desired extension? I mean, k[t]/k itself is not an extension, a field extension that is, which I can only assume is what is meant. I was thinking k(t) at first, but I really don't know. I would really appreciate help with this one, thanks.

The field extension you want is (I omitted the /k for notational convenience). First, you need to see that is not a square. Then observe that the polynomial has derivative (since you have characteristic 2). Therefore is not separable. It is easy to see that is the minimal polynomial for a root in the extension . To conclude, is an inseparable minimal polynomial for , so the extension is inseparable.

Re: Inseparable extension of rational function field

Thank you! Getting really abstract here...