Polynomial Algorithms for Primality Testing in Algebraic Number Fieldswith Class Number 1

We obtain analogues of the Miller and Euler primality criteria in algebraic number fields of class number 1. Based on the obtained primality criteria, we present new efficient probabilistic algorithms for testing primality in algebraic number fields. Assuming that the extended Riemann hypothesis is valid, we obtain efficient deterministic polynomial time algorithms for testing primality in algebraic number fields. We discuss and compare the obtained primality tests with known results.