The determination of all imaginary, quartic, abelian number fields with class number

Author:
Bennett Setzer

Journal:
Math. Comp. **35** (1980), 1383-1386

MSC:
Primary 12A30; Secondary 12A50

DOI:
https://doi.org/10.1090/S0025-5718-1980-0583516-2

MathSciNet review:
583516

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Abstract: In this paper, it is proved that there are just seven imaginary number fields, quartic cyclic over the rational field, and having class number 1. These are the quartic, cyclic imaginary subfields of the cyclotomic fields generated by the *f*th roots of unity, where *f* is 16 or is a prime less than 100. This completes the list of imaginary, quartic, abelian number fields with class number 1. There are 54 such fields, with maximal conductor 67.163.

**[1]**E. BROWN & C. J. PARRY, "The imaginary bicyclic biquadratic fields with class number 1,"*J. Reine Angew. Math.*, v. 266, 1974, pp. 118-120. MR**0340219 (49:4974)****[2]**HELMUT HASSE,*Über die Klassenzahl abelscher Zahlkorper*, Akademie-Verlag, Berlin, 1952. MR**0049239 (14:141a)****[3]**H. L. MONTGOMERY & P. J. WEINBERGER, "Notes on small class numbers,"*Acta Arith.*, v. 24, 1974, pp. 529-542. MR**0357373 (50:9841)****[4]**HAROLD STARK, "On complex quadratic fields with class number two,"*Math. Comp.*, v. 29, 1975, pp. 289-302. MR**0369313 (51:5548)****[5]**KÔJI UCHIDA, "Imaginary abelian number fields with class number 1,"*Tôhoku Math. J.*(2), v. 24, 1972, pp. 487-499. MR**0321904 (48:269)**

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DOI:
https://doi.org/10.1090/S0025-5718-1980-0583516-2

Article copyright:
© Copyright 1980
American Mathematical Society