List of number fields with class number one

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

Contents

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 ):

(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 9 following negative values of d:

(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]

  • x3x2 2x + 1 (discriminant 49)
  • x3 3x 1 (discriminant 81)
  • x3x2 3x + 1 (discriminant 148)
  • x3x2 4x 1 (discriminant 169)
  • x3 4x 1 (discriminant 229)
  • x3x2 4x + 3 (discriminant 257)
  • x3x2 4x + 2 (discriminant 316)
  • x3x2 4x + 1 (discriminant 321)
  • x3x2 6x + 7 (discriminant 361)
  • x3x2 5x 1 (discriminant 404)
  • x3x2 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 real root of the polynomial for 23 is the reciprocal of the plastic ratio (negated), while that for 31 is the reciprocal of the supergolden ratio. The polynomials defining the complex cubic fields that have class number one and discriminant greater than 500 are: [5]

  • x3x2 + 1 (discriminant 23)
  • x3 + x 1 (discriminant 31)
  • x3x2 + x + 1 (discriminant 44)
  • x3 + 2x 1 (discriminant 59)
  • x3 2x 2 (discriminant 76)
  • x3x2 + x 2 (discriminant 83)
  • x3x2 + 2x + 1 (discriminant 87)
  • x3x 2 (discriminant 104)
  • x3x2 + 3x 2 (discriminant 107)
  • x3 2 (discriminant 108)
  • x3x2 2 (discriminant 116)
  • x3 + 3x 1 (discriminant 135)
  • x3x2 + x + 2 (discriminant 139)
  • x3 + 2x 2 (discriminant 140)
  • x3x2 2x 2 (discriminant 152)
  • x3x2x + 3 (discriminant 172)
  • x3x2 + 2x 3 (discriminant 175)
  • x3x2 + 4x 1 (discriminant 199)
  • x3x2 + 2x + 2 (discriminant 200)
  • x3x2 + x 3 (discriminant 204)
  • x3 2x 3 (discriminant 211)
  • x3x2 + 4x 2 (discriminant 212)
  • x3 + 3x 2 (discriminant 216)
  • x3x2 + 3 (discriminant 231)
  • x3x 3 (discriminant 239)
  • x3 3 (discriminant 243)
  • x3 + x 6 (discriminant 244)
  • x3 + x 3 (discriminant 247)
  • x3x2 3 (discriminant 255)
  • x3x2 3x + 5 (discriminant 268)
  • x3x2 3x 3 (discriminant 300)
  • x3x2 + 3x + 2 (discriminant 307)
  • x3 3x 4 (discriminant 324)
  • x3x2 2x 3 (discriminant 327)
  • x3x2 + 4x + 1 (discriminant 335)
  • x3x2x + 4 (discriminant 339)
  • x3 + 3x 3 (discriminant 351)
  • x3x2 + x + 7 (discriminant 356)
  • x3 + 4x 2 (discriminant 364)
  • x3x2 + 2x + 3 (discriminant 367)
  • x3x2 + x 4 (discriminant 379)
  • x3x2 + 5x 2 (discriminant 411)
  • x3 4x 5 (discriminant 419)
  • x3x2 + 8 (discriminant 424)
  • x3x 8 (discriminant 431)
  • x3 + x 4 (discriminant 436)
  • x3x2 2x + 5 (discriminant 439)
  • x3 + 2x 8 (discriminant 440)
  • x3x2 5x + 8 (discriminant 451)
  • x3 + 3x 8 (discriminant 459)
  • x3x2 + 5x 3 (discriminant 460)
  • x3 5x 6 (discriminant 472)
  • x3x2 + 4x + 2 (discriminant 484)
  • x3x2 + 3x + 3 (discriminant 492)
  • x3 + 4x 3 (discriminant 499)

Cyclotomic fields

The following is a complete list of thirty n for which the field n) has class number 1: [6] [7]

(Note that values of n congruent to 2 modulo 4 are redundant since 2n) = n) when n is odd.)

On the other hand, the maximal real subfields (cos(2π/2n)) of the 2-power cyclotomic fields 2n) (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 [9] 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, [10] and later improved this bound to 109. [11] These fields are the n-th layers of the cyclotomic 2-extension of . Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic 3-extension of have no prime factor less than 104. [12] Coates has raised the question of whether, for all primes p, every layer of the cyclotomic p-extension of has class number 1. [13]

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. [14] 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. [15] 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. [16]

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. [17] Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1. [18]

See also

Notes

  1. 1 2 3 4 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. S2CID   9088028. Zbl   1152.11328.
  3. H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer Verlag (1993), Appendix B2, p.507
  4. Cohen, H.; Lenstra, H. W. (1984). "Heuristics on class groups of number fields". Number Theory Noordwijkerhout 1983. Vol. 1068. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 33–62. doi: 10.1007/bfb0099440 . ISBN   978-3-540-13356-8 . Retrieved 2025-04-20.
  5. 1 2 Tables available at Pari source code
  6. Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Graduate Texts in Mathematics. Vol. 83 (2nd ed.). Springer-Verlag. Theorem 11.1. ISBN   0-387-94762-0. Zbl   0966.11047.
  7. Sloane, N. J. A. (ed.). "SequenceA005848(Cyclotomic fields with class number 1 (or with unique factorization).)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2024-03-20.
  8. Miller, John C. (2014). "Class numbers of totally real fields and applications to the Weber class number problem" (PDF). Acta Arithmetica. 164 (4): 381–397. doi: 10.4064/aa164-4-4 . ISSN   0065-1036 . Retrieved 2025-04-20.
  9. Weber, H. (1886). "Theorie der Abel'schen Zahlkörper" (PDF). Acta Mathematica. 8 (0): 193–263. doi: 10.1007/BF02417089 . ISSN   0001-5962 . Retrieved 2025-04-22.
  10. 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. S2CID   31421633. Zbl   1189.11033.
  11. 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. S2CID   121397082. Zbl   1226.11119.
  12. 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.
  13. Coates, John (2011). The enigmatic Tate-Shafarevich group. Proceedings of the Fifth International Congress of Chinese Mathematicians. Beijing, China.
  14. 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 , S2CID   119482000
  15. 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, S2CID   119348804
  16. Murty, V. Kumar (2001), "Class numbers of CM-fields with solvable normal closure", Compositio Mathematica , 127 (3): 273–287, doi: 10.1023/A:1017589432526
  17. 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
  18. 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