MATHEMATICS OF COMPUTATION Volume 70, Number 234, Pages 845–851 S 0025-5718(00)01232-1 Article electronically published on July 13, 2000 TENTH DEGREE NUMBER FIELDS WITH QUINTIC FIELDS HAVING ONE REAL PLACE SCHEHRAZAD SELMANE Abstract. In this paper, we enumerate all number fields of degree 10 of dis- criminant smaller than 10 11 in absolute value containing a quintic field having one real place. For each one of the 21509 (resp. 18167) found fields of signa- ture (0, 5) (resp. (2, 4)) the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding rela- tive discriminant, the corresponding polynomial over Q, and the Galois group of the Galois closure are given. In a supplementary section, we give the first coincidence of discriminant of 19 (resp. 20) nonisomorphic fields of signature (0, 5) (resp. (2, 4)). 1. Introduction In this paper, we enumerate all number fields of degree 10 of discriminant smaller than 10 11 in absolute value containing a quintic field having one real place. The list corresponding to the signature (0, 5) (resp. (2, 4)) contains 21509 (resp. 18167) fields. For each field in the given two lists, the field discriminant, the quintic field discriminant, a polynomial defining the relative quadratic extension, the corresponding relative discriminant, the corresponding polynomial over Q, and the Galois group of the Galois closure are given. To establish these lists, we have followed, without major modification, the method of [4, 7]; Section 2 of this paper contains a brief description of the method used. We describe the results in the third section. The existence of several noniso- morphic fields, of composite fields, of unramified extensions, of Hilbert class fields as well as other characteristics, have been examined in more detail. Finally, in a supplementary section, we give the first coincidence of a discriminant of 19 (resp. (2, 4)) nonisomorphic fields of signature (0, 5) (resp. (2, 4)) with a value of its discriminants out of the originally chosen bound. 2. The method If L is a number field of degree n, we denote by ϑ L its ring of integers, by d L its discriminant, and by h L its class number. For β ∈ L, we denote the corresponding conjugates by β (1) , ..., β (n) and set T 2 (β)= ∑ n i=1 |β (i) | 2 . To establish the lists of all number fields of degree 10 over Q and of field dis- criminant smaller than 10 11 in absolute value containing a quintic subfield having one real place, we have followed without major modification the method of explicit Received by the editor November 3, 1998 and, in revised form, April 27, 1999. 2000 Mathematics Subject Classification. Primary 11R11, 11R29, 11Y40. Key words and phrases. Quintic fields, relative quadratic extensions, discriminant. c 2000 American Mathematical Society 845 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use