The Ring-LWE problem, which is expected to be difficult to calculate even with a quantum computer, is drawing attention as the development technology of the quantum computer advances. At present, the Ring-LWE problem is often constructed on a circle field. The reason is that there is a basis suitable for high-speed computation, and it can be used for the construction of efficient cryptographic protocols. On the other hand, Arita-Handa et al. clarified that there exists a basis suitable for high-speed operations in algebraic field algebra called decomposition field, and construct homomorphic encryption based on Ring-LWE problem on decomposition field. It has been shown that it is possible to efficiently process more plain texts collectively than in the case of using the circle field. There are also proposals for research projects that search for an optimal algebraic field rather than a circle field depending on the application and construct a Ring-LWE-based cryptographic protocol on the algebraic field. Therefore, it is important to study the existence of a basis suitable for high-speed operations in algebraic fields other than the circle field and the decomposition field. In this paper, we study the basis of the subfields of the circle field that can be operated at high speed suitable for the Ring-LWE problem.