The length of the group algebra of the dihedral group of order $2^k$

In this paper, the length of the group algebra of a dihedral group in the modular case is computed under the assumption that the order of the group is a power of two. Various methods for studying the length of a group algebra in the modular case are considered. It is proved that the length of the group algebra of a dihedral group of order $2^{k+1} $ over an arbitrary field of characteristic $2$ is equal to $2^{k}$.