Automorphisms and extensions of \(k((t))\)
Let \(k\) be a field of characteristic \(p\) and let \(K=k((t))\) be the power series field over \(k\). For an automorphism \(\sigma\) of \(K\) over \(k\), let \(i(\sigma)\) be the \(K\)-valuation of \((\sigma t-t)/t\) and define \(i_ m\) to be \(i\) of the \(p^ m\)-th power of \(\sigma\). The author studies the properties of the sequence of integers \(i_ m\). To explain the results, let \(a\) denote \(i_ 0\) minus the greatest integer in \((i_ 0/p)\); let \(b=(i_ 1-i_ 0)/p\); and let \(K(\sigma)\) denote the abelian extension \(L\) of \(K\) maximal with respect to the property that any lift of \(\sigma\) to \(K^{sep}\) preserves \(L\) setwise and conjugation by it on \(\text{Gal}(L/K)\) is the identity. The author's first theorem is that if \(k\) is a finite field of \(p^ f\) elements and \(\sigma\) is such that \(i_ 0\) is prime to \(p\) and \(i_ 1\) is less than \((p^ 2-p+1)i_ 0\) then the \(p\)-rank of \(G=\text{Gal}(K(\sigma)/K)\) (the dimension over \(\mathbb{F}_ p\) of \(G/G^ p\)) is at most \(f(a+b-1)\). He calculates the \(p\)-rank of \(G\) via a filtration by ramification subgroups, and obtains his main theorem: with \(i_ 0\) and \(i_ 1\) as in the preceding theorem, and with also \(i_ 1\) greater than \(p^ 2(a+b-1-p)/(p-1)\) then \(i_ m-i_{m-1}\leq p^ m(a+b-1)\). As a corollary, he obtains that if \(i_ 0=1\) and \(i_ 1=1+bp\) with \(b<p-1\) then \(i_ m=1+bp+bp^ 2+\dots +bp^ m\).
scientific article; zbMATH DE number 96567