This paper generalizes the work of B. Erez [“The Galois structure of the trace form in extensions of odd prime degree”, J. Algebra 118, No. 2, 438–446 (1988; Zbl 0663.12015)] in constructing self-dual integral normal bases for the square-root of the inverse different \(A_{L|K}\) in certain odd abelian extensions of \(p\)-adic local fields. Starting with \(K\) a finite unramified extension of \(\mathbb{Q}_p\), let \(K_{p,n}\) be the field of \([p^n]\)-division points associated to the Lubin-Tate formal group associated to the uniformizer \(p\) in \(K\). Then the construction in this paper applies to \(p\)-cyclic extensions \(M|K\) contained in \(K_{p,2}\). Such extensions are weakly ramified, as is necessary and sufficient for the existence of a self-dual integral normal basis generator for \(A_{M|K}\).
The argument begins with the construction of explicit Kummer generators for the extension \(K_{p,2}|K_{p,1}\), using Dwork’s exponential power series to define some special elements in the base field, and then local class field theory to show that their \(p\)th roots generate \(K_{p,2}\). For any \(p\)-cyclic extension \(M|K\) as above, this description of the generators allows for the explicit construction of an element of \(A_{M|K}\) that is self-dual with respect to the trace form \(T_{M|K}\). As the author explains in Lemma 8, any such self-dual element is automatically an integral normal basis generator for \(A_{M|K}\). In the case \(K=\mathbb{Q}_p\), the resulting self-dual normal basis is the same as that obtained by Erez in the paper cited above.