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The degree of a particular purely inseparable extension

  1. Mar 28, 2010 #1

    eof

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    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):

    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...).
     
  2. jcsd
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