List of number fields with class number one

This is an incomplete list of number fields with class number 1.

It is believed that there are infinitely many such number fields, but this has not been proven.[1]

Definition

The class number of a number field is by definition the order of the ideal class group of its ring of integers.

Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain (and thus a unique factorization domain). The fundamental theorem of arithmetic says that Q has class number 1.

Quadratic number fields

These are of the form K = Q(d), for a square-free integer d.

Real quadratic fields

K is called real quadratic if d > 0. K has class number 1 for the following values of d (sequence A003172 in the OEIS):

  • 2*, 3, 5*, 6, 7, 11, 13*, 14, 17*, 19, 21, 22, 23, 29*, 31, 33, 37*, 38, 41*, 43, 46, 47, 53*, 57, 59, 61*, 62, 67, 69, 71, 73*, 77, 83, 86, 89*, 93, 94, 97*, ...[1][2]

(complete until d = 100)

*: The narrow class number is also 1 (see related sequence A003655 in OEIS).

Despite what would appear to be the case for these small values, not all prime numbers that are congruent to 1 modulo 4 appear on this list, notably the fields Q(d) for d = 229 and d = 257 both have class number greater than 1 (in fact equal to 3 in both cases).[3] The density of such primes for which Q(d) does have class number 1 is conjectured to be nonzero, and in fact close to 76%,[4] however it is not even known whether there are infinitely many real quadratic fields with class number 1.[1]

Imaginary quadratic fields

K has class number 1 exactly for the following negative values of d:

  • 1, 2, 3, 7, 11, 19, 43, 67, 163.[1]

(By definition, these also all have narrow class number 1.)

Cubic fields

Totally real cubic field

The first 60 totally real cubic fields (ordered by discriminant) have class number one. In other words, all cubic fields of discriminant between 0 and 1944 (inclusively) have class number one. The next totally real cubic field (of discriminant 1957) has class number two. The polynomials defining the totally real cubic fields that have discriminants less than 500 with class number one are:[5]

  • x3 x2 2x + 1 (discriminant 49)
  • x3 3x 1 (discriminant 81)
  • x3 x2 3x + 1 (discriminant 148)
  • x3 x2 4x 1 (discriminant 169)
  • x3 4x 1 (discriminant 229)
  • x3 x2 4x + 3 (discriminant 257)
  • x3 x2 4x + 2 (discriminant 316)
  • x3 x2 4x + 1 (discriminant 321)
  • x3 x2 6x + 7 (discriminant 361)
  • x3 x2 5x 1 (discriminant 404)
  • x3 x2 5x + 4 (discriminant 469)
  • x3 5x 1 (discriminant 473)

Complex cubic field

All complex cubic fields with discriminant greater than 500 have class number one, except the fields with discriminants 283, 331 and 491 which have class number 2. The polynomials defining the complex cubic fields that have class number one and discriminant greater than 500 are:[5]

  • x3 x2 + 1 (discriminant 23)
  • x3 + x 1 (discriminant 31)
  • x3 x2 + x + 1 (discriminant 44)
  • x3 + 2x 1 (discriminant 59)
  • x3 2x 2 (discriminant 76)
  • x3 x2 + x 2 (discriminant 83)
  • x3 x2 + 2x + 1 (discriminant 87)
  • x3 x 2 (discriminant 104)
  • x3 x2 + 3x 2 (discriminant 107)
  • x3 2 (discriminant 108)
  • x3 x2 2 (discriminant 116)
  • x3 + 3x 1 (discriminant 135)
  • x3 x2 + x + 2 (discriminant 139)
  • x3 + 2x 2 (discriminant 140)
  • x3 x2 2x 2 (discriminant 152)
  • x3 x2 x + 3 (discriminant 172)
  • x3 x2 + 2x 3 (discriminant 175)
  • x3 x2 + 4x 1 (discriminant 199)
  • x3 x2 + 2x + 2 (discriminant 200)
  • x3 x2 + x 3 (discriminant 204)
  • x3 2x 3 (discriminant 211)
  • x3 x2 + 4x 2 (discriminant 212)
  • x3 + 3x 2 (discriminant 216)
  • x3 x2 + 3 (discriminant 231)
  • x3 x 3 (discriminant 239)
  • x3 3 (discriminant 243)
  • x3 + x 6 (discriminant 244)
  • x3 + x 3 (discriminant 247)
  • x3 x2 3 (discriminant 255)
  • x3 x2 3x + 5 (discriminant 268)
  • x3 x2 3x 3 (discriminant 300)
  • x3 x2 + 3x + 2 (discriminant 307)
  • x3 3x 4 (discriminant 324)
  • x3 x2 2x 3 (discriminant 327)
  • x3 x2 + 4x + 1 (discriminant 335)
  • x3 x2 x + 4 (discriminant 339)
  • x3 + 3x 3 (discriminant 351)
  • x3 x2 + x + 7 (discriminant 356)
  • x3 + 4x 2 (discriminant 364)
  • x3 x2 + 2x + 3 (discriminant 367)
  • x3 x2 + x 4 (discriminant 379)
  • x3 x2 + 5x 2 (discriminant 411)
  • x3 4x 5 (discriminant 419)
  • x3 x2 + 8 (discriminant 424)
  • x3 x 8 (discriminant 431)
  • x3 + x 4 (discriminant 436)
  • x3 x2 2x + 5 (discriminant 439)
  • x3 + 2x 8 (discriminant 440)
  • x3 x2 5x + 8 (discriminant 451)
  • x3 + 3x 8 (discriminant 459)
  • x3 x2 + 5x 3 (discriminant 460)
  • x3 5x 6 (discriminant 472)
  • x3 x2 + 4x + 2 (discriminant 484)
  • x3 x2 + 3x + 3 (discriminant 492)
  • x3 + 4x 3 (discriminant 499)

Cyclotomic fields

The following is a complete list of n for which the field Qn) has class number 1:[6]

  • 1 through 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 44, 45, 48, 50, 54, 60, 66, 70, 84, 90.[7]

On the other hand, the maximal real subfields Q(cos(2π/2n)) of the 2-power cyclotomic fields Q2n) (where n is a positive integer) are known to have class number 1 for n≤8,[8] and it is conjectured that they have class number 1 for all n. Weber showed that these fields have odd class number. In 2009, Fukuda and Komatsu showed that the class numbers of these fields have no prime factor less than 107,[9] and later improved this bound to 109.[10] These fields are the n-th layers of the cyclotomic Z2-extension of Q. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic Z3-extension of Q have no prime factor less than 104.[11] Coates has raised the question of whether, for all primes p, every layer of the cyclotomic Zp-extension of Q has class number 1.

CM fields

Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1.[12] He showed that there are finitely many of a fixed degree. Shortly thereafter, Andrew Odlyzko showed that there are only finitely many Galois CM fields of class number 1.[13] In 2001, V. Kumar Murty showed that of all CM fields whose Galois closure has solvable Galois group, only finitely many have class number 1.[14]

A complete list of the 172 abelian CM fields of class number 1 was determined in the early 1990s by Ken Yamamura and is available on pages 915–919 of his article on the subject.[15] Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.[16]

See also

Notes

  1. Chapter I, section 6, p. 37 of Neukirch 1999
  2. Dembélé, Lassina (2005). "Explicit computations of Hilbert modular forms on " (PDF). Exp. Math. 14 (4): 457–466. doi:10.1080/10586458.2005.10128939. ISSN 1058-6458. Zbl 1152.11328.
  3. H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer Verlag (1993), Appendix B2, p.507
  4. H. Cohen and H. W. Lenstra, Heuristics on class groups of number fields, Number Theory, Noordwijkerhout 1983, Proc. 13th Journées Arithmétiques, ed. H. Jager, Lect. Notes in Math. 1068, Springer-Verlag, 1984, pp. 33—62
  5. Tables available at Pari source code
  6. Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. 83 (2nd ed.). Springer-Verlag. Theorem 11.1. ISBN 0-387-94762-0. Zbl 0966.11047.
  7. Note that values of n congruent to 2 modulo 4 are redundant since Q2n) = Qn) when n is odd.
  8. J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, https://arxiv.org/abs/1405.1094
  9. Fukuda, Takashi; Komatsu, Keiichi (2009). "Weber's class number problem in the cyclotomic -extension of ". Exp. Math. 18 (2): 213–222. doi:10.1080/10586458.2009.10128896. ISSN 1058-6458. MR 2549691. Zbl 1189.11033.
  10. Fukuda, Takashi; Komatsu, Keiichi (2011). "Weber's class number problem in the cyclotomic -extension of III". Int. J. Number Theory. 7 (6): 1627–1635. doi:10.1142/S1793042111004782. ISSN 1793-7310. MR 2835816. Zbl 1226.11119.
  11. Morisawa, Takayuki (2009). "A class number problem in the cyclotomic -extension of ". Tokyo J. Math. 32 (2): 549–558. doi:10.3836/tjm/1264170249. ISSN 0387-3870. MR 2589962. Zbl 1205.11116.
  12. Stark, Harold (1974), "Some effective cases of the Brauer–Siegel theorem", Inventiones Mathematicae, 23 (2): 135–152, Bibcode:1974InMat..23..135S, doi:10.1007/bf01405166, hdl:10338.dmlcz/120573
  13. Odlyzko, Andrew (1975), "Some analytic estimates of class numbers and discriminants", Inventiones Mathematicae, 29 (3): 275–286, Bibcode:1975InMat..29..275O, doi:10.1007/bf01389854
  14. Murty, V. Kumar (2001), "Class numbers of CM-fields with solvable normal closure", Compositio Mathematica, 127 (3): 273–287, doi:10.1023/A:1017589432526
  15. Yamamura, Ken (1994), "The determination of the imaginary abelian number fields with class number one", Mathematics of Computation, 62 (206): 899–921, Bibcode:1994MaCom..62..899Y, doi:10.2307/2153549, JSTOR 2153549
  16. Louboutin, Stéphane; Okazaki, Ryotaro (1994), "Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one", Acta Arithmetica, 67 (1): 47–62, doi:10.4064/aa-67-1-47-62

References

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