Finite \(p\)-groups and \(k((t))\). (Q765693)

Let \(p\) be a prime, \(G\) a finite group of order \(p^n\), \(k\) a field of characteristic \(p>0\) and \(K=k((t))\) the field of finitely tailed Laurent series over \(k\). Then there exists a unit \(u\in k[[t]]\) such that \(K/k((ut^{p^n}))\) is Galois with Galois group isomorphic to \(G\). The authors investigate the relationship between \(u\) and \(G\). In particular, they prove that if \(u_1\) and \(u_2\) are units that are sufficiently close in the \(t\)-adic topology then \(\text{Gal}(K/L_1)\cong\text{Gal}(K/L_2)\) where \(L_i=k((u_it^{p^n}))\) for \(i=1,2\). The authors raise interesting questions, asking whether properties of the coefficients of \(u\) determine algebraic properties of \(G\), for example whether \(G\) is cyclic or Abelian. Conversely, do properties of \(G\) determine the sequence of coefficients of \(u\)? Note, when \(k\) is a finite field these investigations relate to results about finite \(p\)-groups embedding in the so-called Nottingham group.