Probabilistic and deterministic analogues of the Miller – Rabin algorithm for ideals of rings of integer algebraic elements of finite extensions of the field 

In this paper, we obtained the primality criteria for ideals of rings of integer algebraic elements of finite extensions of the field Q, which are analogues of Miller and Euler’s primality criteria for rings of integers. Also advanced analogues of these criteria were obtained, assuming the extended Riemann hypothesis. Arithmetic and modular operations for ideals of rings of integer algebraic elements of finite extensions of the field Q were elaborated. Using these criteria, the polynomial probabilistic and deterministic algorithms for the primality testing in rings of integer algebraic elements of finite extensions of the field Q were offered.