THE RING GENERATED BY THE Q-TH POWERS OF THE INTEGERS OF AN ALGEBRRAIC NUMBER FIELD, Q BEING A PRIME
Abstract
The following Theorem is proved: Let K be an algebraic number field of finite degree n over the rationals and let J (K) be its ring of integers. Let J sub q (K) be the additive group generated by the qth powers of the elements of J (K). Suppose q is a prime number, then J sub q (K) is equal to J (K), if and only if neither of the following circumstances obtains in K: (1) q is ramified in K, or (2) q is expressible in the form (p to the r power -1)/ (p to the d power -1), where p is a prime, r and d are positive integers, and p has in J (K) a prime ideal factor of degree r.
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1961
- Accession Number
- AD0264335
Entities
People
- Paul T. Bateman
- Rosemarie S. Stemmler
Organizations
- University of Illinois Urbana–Champaign