Class number parity for the $p$th cyclotomic field
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Abstract:
We study the parity of the class number of the pth cyclotomic field for p prime. By analytic methods we derive a parity criterion in terms of polynomials over the field of 2 elements. The conjecture that the class number is odd for p a prime of the form $2q + 1$, with q prime, is proved in special cases, and a heuristic argument is given in favor of the conjecture. An implementation of the criterion on a computer shows that no small counterexamples to the conjecture exist.References
- Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367. MR 148632, DOI 10.1090/S0025-5718-1962-0148632-7
- Gary Cornell and Michael I. Rosen, The $\ell$-rank of the real class group of cyclotomic fields, Compositio Math. 53 (1984), no. 2, 133–141. MR 766293
- Daniel Davis, Computing the number of totally positive circular units which are squares, J. Number Theory 10 (1978), no. 1, 1–9. MR 476695, DOI 10.1016/0022-314X(78)90002-1
- Dennis R. Estes, On the parity of the class number of the field of $q$th roots of unity, Rocky Mountain J. Math. 19 (1989), no. 3, 675–682. Quadratic forms and real algebraic geometry (Corvallis, OR, 1986). MR 1043240, DOI 10.1216/RMJ-1989-19-3-675
- Ke Qin Feng, An elementary criterion on parity of class number of cyclic number field, Sci. Sinica Ser. A 25 (1982), no. 10, 1032–1041. MR 690929
- Georges Gras, Parité du nombre de classes et unités cyclotomiques, Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) Astérisque, No. 24–25, Soc. Math. France, Paris, 1975, pp. 37–45 (French). D’après un travail en commun avec Marie-Nicole Gras. MR 0382224
- Georges Gras, Nombre de $\varphi$-classes invariantes. Application aux classes des corps abéliens, Bull. Soc. Math. France 106 (1978), no. 4, 337–364 (French). MR 518043
- Jürgen Hurrelbrink, Class numbers, units and $K_2$, Algebraic $K$-theory: connections with geometry and topology (Lake Louise, AB, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 87–102. MR 1045846, DOI 10.1007/978-94-009-2399-7_{4}
- Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
- Serge Lang, Units and class groups in number theory and algebraic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 253–316. MR 648522, DOI 10.1090/S0273-0979-1982-14997-7
- Olga Taussky, Unimodular integral circulants, Math. Z. 63 (1955), 286–289. MR 72890, DOI 10.1007/BF01187938
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Math. Comp. 63 (1994), 773-784
- MSC: Primary 11R29; Secondary 11R18
- DOI: https://doi.org/10.1090/S0025-5718-1994-1242060-X
- MathSciNet review: 1242060