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 9 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 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]

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 thirty n for which the field Q(ζ_n) has class number 1:^[6]^[7]

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.

(Note that values of n congruent to 2 modulo 4 are redundant since Q(ζ_2n) = Q(ζ_n) when n is odd.)

On the other hand, the maximal real subfields Q(cos(2π/2ⁿ)) of the 2-power cyclotomic fields Q(ζ_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 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 Z_p-extension of Q has class number 1.^{[citation needed]}

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]

Notes

^ ^a ^b ^c ^d Chapter I, section 6, p. 37 of Neukirch 1999
^ Dembélé, Lassina (2005). "Explicit computations of Hilbert modular forms on $\mathbb {Q} ({\sqrt {5}})$ " (PDF). Exp. Math. 14 (4): 457–466. doi:10.1080/10586458.2005.10128939. ISSN 1058-6458. S2CID 9088028. Zbl 1152.11328.
^ H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer Verlag (1993), Appendix B2, p.507
^ 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
^ ^a ^b Tables available at Pari source code
^ 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.
^ Sloane, N. J. A. (ed.). "Sequence A005848 (Cyclotomic fields with class number 1 (or with unique factorization).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-20.
^ J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, https://arxiv.org/abs/1405.1094
^ Fukuda, Takashi; Komatsu, Keiichi (2009). "Weber's class number problem in the cyclotomic $\mathbb {Z} _{2}$ -extension of $\mathbb {Q}$ ". Exp. Math. 18 (2): 213–222. doi:10.1080/10586458.2009.10128896. ISSN 1058-6458. MR 2549691. S2CID 31421633. Zbl 1189.11033.
^ Fukuda, Takashi; Komatsu, Keiichi (2011). "Weber's class number problem in the cyclotomic $\mathbb {Z} _{2}$ -extension of $\mathbb {Q}$ III". Int. J. Number Theory. 7 (6): 1627–1635. doi:10.1142/S1793042111004782. ISSN 1793-7310. MR 2835816. S2CID 121397082. Zbl 1226.11119.
^ Morisawa, Takayuki (2009). "A class number problem in the cyclotomic $\mathbb {Z} _{3}$ -extension of $\mathbb {Q}$ ". Tokyo J. Math. 32 (2): 549–558. doi:10.3836/tjm/1264170249. ISSN 0387-3870. MR 2589962. Zbl 1205.11116.
^ 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
^ 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
^ Murty, V. Kumar (2001), "Class numbers of CM-fields with solvable normal closure", Compositio Mathematica, 127 (3): 273–287, doi:10.1023/A:1017589432526
^ 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
^ 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

Neukirch, Jürgen (1999). Algebraische Zahlentheorie. Grundlehren der mathematischen Wissenschaften. Vol. 322. Berlin: Springer-Verlag. ISBN 978-3-540-65399-8. MR 1697859. Zbl 0956.11021.

[neu-1] Chapter I, section 6, p. 37 of Neukirch 1999

[2] Dembélé, Lassina (2005). "Explicit computations of Hilbert modular forms on $\mathbb {Q} ({\sqrt {5}})$ " (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] 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

[pari-5] 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.). "Sequence A005848 (Cyclotomic fields with class number 1 (or with unique factorization).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-20.

[8] J. C. Miller, Class numbers of totally real fields and applications to the Weber class number problem, https://arxiv.org/abs/1405.1094

[Exp2009-9] Fukuda, Takashi; Komatsu, Keiichi (2009). "Weber's class number problem in the cyclotomic $\mathbb {Z} _{2}$ -extension of $\mathbb {Q}$ ". Exp. Math. 18 (2): 213–222. doi:10.1080/10586458.2009.10128896. ISSN 1058-6458. MR 2549691. S2CID 31421633. Zbl 1189.11033.

[10] Fukuda, Takashi; Komatsu, Keiichi (2011). "Weber's class number problem in the cyclotomic $\mathbb {Z} _{2}$ -extension of $\mathbb {Q}$ III". Int. J. Number Theory. 7 (6): 1627–1635. doi:10.1142/S1793042111004782. ISSN 1793-7310. MR 2835816. S2CID 121397082. Zbl 1226.11119.

[11] Morisawa, Takayuki (2009). "A class number problem in the cyclotomic $\mathbb {Z} _{3}$ -extension of $\mathbb {Q}$ ". 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, S2CID 119482000

[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, S2CID 119348804

[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

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