The structure of formal modules as Galois modules in cyclic unramified $p$-extensions

The structure of the formal module $F(\mathfrak{M})$ for a chain of finite extensions $M/L/K$, where $M/L$ is an unramified $p$-extension, is studied. The triviality of the first Galois cohomology of a formal module for an unramified extension for any degree of a prime ideal is shown. The presentation of the investigated formal module is constructed in terms of generators and relations. As an application of the main result, the structure of a formal module for generalized Lubin–Tate formal groups is obtained.