The associated orders of rings of integers in Lubin-Tate division fields over the \(p\)-adic number field. (English) Zbl 0816.11061

The associated order of a Galois extension \(L/K\) is the subset \(\{\lambda\in K[\Gamma ]\mid \lambda{\mathcal O}_ L \subseteq{\mathcal O}_ L\}\) of the group ring \(K[\Gamma ]\), where \(\Gamma= \text{Gal} (L/K)\), \({\mathcal O}_ L\) is the valuation ring (if \(L\) is local) or the ring of integers (if \(L\) is global). Let \(\mathbb{Q}_ p\) denote the \(p\)-adic number field, \(\mathbb{Q}^ n_{p,\pi}\) the division field of level \(n\) and uniformizer \(\pi\) associated to some Lubin-Tate formal group. The authors determine the generators of the associated order of \(L/K\) with \(L= \mathbb{Q}_{p,\pi}^{m+r}\), \(K= \mathbb{Q}^ r_{p,\pi}\). This work is based on the authors’ former work [J. Reine Angew. Math. 434, 205-220 (1993; Zbl 0753.11038)], by adjoining an unramified extension to \(K\) to obtain a relative cyclotomic extension and descending then to \(\mathbb{Q}_ p\).


11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S31 Class field theory; \(p\)-adic formal groups


Zbl 0753.11038