Finite p-groups and k((t))

Let k be a field of characteristic p>0 and let K=k((t)) be the field of Laurent series over k. For each group G of order pn there exist units u∈k〚t〛 such that K/k((utpn)) is Galois with Gal(K/k((utpn)))≅G. We explore the connections between G and u. Among other results, we prove that if both K/k((u1tpn)) and K/k((u2tpn)) are Galois and u1 and u2 are sufficiently close in the t-adic topology, then Gal(K/k((u1tpn)))≅Gal(K/k((u2tpn))).