A005848 Cyclotomic fields with class number 1 (or with unique factorization).

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 48, 60, 84

Offset: 1

Author

N. J. A. Sloane

Keywords

• fini
• nonn
• full
• nice

Comments

Note that if n == 2 (mod 4) Q(zeta_n) is the same field as Q(zeta_{n/2}), so this sequence omits numbers that are 2 mod 4. - Yuval Dekel, Jun 07 2003

Also note that 3 corresponds to Z[omega] (the Eisenstein integers) and 4 corresponds to Z[i] (the Gaussian integers).

Alaca & Williams cite Masley & Montgomery, saying the earlier authors "prove that there are precisely 29 distinct cyclotomic fields" with class number 1 (mentioning the n = 2 mod 4 caveat), and then give this sequence without the initial 1. - Alonso del Arte, Mar 10 2017

References

• Şaban Alaca & Kenneth S. Williams, Introductory Algebraic Number Theory. Cambridge: Cambridge University Press (2004): 343.
• F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 85, 1983.
• J. Myron Masley, Where are the number fields with small class number?, pp. 221-242 of Number Theory Carbondale 1979, Lect. Notes Math. 751 (1982).
• Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 259.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 14.
• L. C. Washington, Introduction to Cyclotomic Fields, Springer, p. 353.

Links

• Michael Baake and Uwe Grimm, A note on shelling, arXiv:math/0203025 [math.MG], 2002-2003.
• E. Bugarin, M. de las Peñas, and D. Frettlöh, Perfect colourings of cyclotomic integers, arXiv:0905.4048 [math.GR], 2009-2012.
• Hendrik W. Lenstra and A. J. van der Poorten, Euclidean number fields 1, Math. Intelligencer 2 (1979): pp. 6-15.
• J. Myron Masley and Hugh L. Montgomery, Cyclotomic fields with unique factorization, Journal für die reine und angewandte Mathematik 286/287 (1976), 248-256.

Crossrefs

Cf. A061653.