Explicit Kummer Generators for Cyclotomic Extensions

A BSTRACT . If p is a prime number congruent to 1 modulo 3 , then we explicitly describe an element of the cyclotomic field Q ( ζ 3 ) whose third root generates the cubic subextension of Q ( ζ 3 p ) / Q ( ζ 3 ) . Similarly, if p is a prime number congruent to 1 modulo 4 , then we explicitly describe an element of the cyclotomic field Q ( ζ 4 ) whose fourth root generates the quartic cyclic subextension of Q ( ζ 4 p ) / Q ( ζ 4 ) . For further number fields we express generators of Kummer extensions inside cyclotomic fields in terms of Gauss sums.