This one step in a long problem (an example in algebraic geometry cast in another language) that I've condensed below (p is the characteristic):(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]\overline{k}[/tex] be an algebraic closure of [tex]k[/tex] and let [tex]L=\overline{k}(Y)[/tex]. Consider the monic polynomial

[tex]

H(x)=x^s-(Y^p+Y^{-t})

[/tex]

in [tex]L[x][/tex]. Write [tex]s=p^ms'[/tex] for some integer [tex]s'[/tex] prime to [tex]p[/tex] and some [tex]m\geq 0[/tex]. Let [tex]x_0[/tex] be a root of [tex]H(x)[/tex] in an algebraic closure [tex]\overline{L}[/tex] of [tex]L[/tex]. Show that the field [tex]L(x_0)[/tex] is an inseparable degree [tex]p^m[/tex] extension of [tex]L(x_1)[/tex] where [tex]x_1=x_0^{p^m}[/tex].

Ok, so proving that the extension is inseparable is trivial. It then follows that the minimal polynomial of [tex]x_0[/tex] over [tex]x_1[/tex] has to be of the form [tex](X-x_0)^{p^n}=X^{p^n}-x_0^{p^n}[/tex]. This shows that [tex]x_0^{p^n}\in L(x_1)[/tex]. Unfortunately, this only gives that [tex]n\leq m[/tex], but it does not show equality. Does anyone have any ideas on how to approach this (I spent about 2 hours yesterday thinking about this...).

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# The degree of a particular purely inseparable extension

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**