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词条 List of number fields with class number one
释义

  1. Definition

  2. Quadratic number fields

     Real quadratic fields   Imaginary quadratic fields  

  3. Cubic fields

  4. Cyclotomic fields

  5. CM fields

  6. See also

  7. Notes

  8. References

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

{{Main|Quadratic number field}}

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

Real quadratic fields

{{Main|Real quadratic field}}

K is called real quadratic if d > 0. K has class number 1 for the following values of d {{OEIS|A003172}}:

  • 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({{radic|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({{radic|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

{{Main|Heegner number}}

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

{{Main|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 discriminants less than 500 with class number one are:

  • 49, 81, 148, 169, 229, 257, 316, 321, 361, 404, 469, 473.

Polynomials defining the first three are respectively:

  • {{nobreak|x3 − x2 − 2x + 1}},
  • {{nobreak|x3 − 3x − 1}},
  • {{nobreak|x3 − x2 − 3x + 1}}.

The first 30 complex cubic fields (ordered by discriminant) have class number one. These are the cubic fields of discriminant between 0 and −268 (inclusively). The next complex cubic field (of discriminant −283) has class number two. The negative discriminants greater than -150 with class number one are:

  • −23, −31, −44, −59, −76, −83, −87, −104, −107, −108, −116, −135, −139, −140.

Polynomials defining the first three are respectively:

  • {{nobreak|x3 − x2 + 1}},
  • {{nobreak|x3 + x − 1}},
  • {{nobreak|x3 − x2 + x + 1}}.[5]

Cyclotomic fields

{{Main|Cyclotomic field}}

The following is a complete list of n for which the field Q(ζn) 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 Q(ζ2n) (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.{{Citation needed|date=January 2016}}

CM fields

{{Main|CM field}}

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

  • Class number problem
  • Class number formula

Notes

1. ^1 2 3 Chapter I, section 6, p. 37 of {{harvnb|Neukirch|1999}}
2. ^{{cite journal | zbl=1152.11328 | last=Dembélé | first=Lassina | title=Explicit computations of Hilbert modular forms on | journal=Exp. Math. | volume=14 | number=4 | pages=457–466 | year=2005 | issn=1058-6458 | url=http://www.emis.de/journals/EM/expmath/volumes/14/14.4/Dembele.pdf | doi=10.1080/10586458.2005.10128939}}
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
6. ^{{cite book | last=Washington | first=Lawrence C. | authorlink=Lawrence C. Washington | title=Introduction to Cyclotomic Fields | publisher=Springer-Verlag | isbn=0-387-94762-0 | zbl=0966.11047| edition=2nd | series=Graduate Texts in Mathematics | volume=83 | year=1997 | at=Theorem 11.1 }}
7. ^Note that values of n congruent to 2 modulo 4 are redundant since Q(ζ2n) = Q(ζn) 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. ^{{cite journal | zbl=1189.11033 | last1=Fukuda | first1=Takashi | last2=Komatsu | first2=Keiichi | title=Weber's class number problem in the cyclotomic -extension of | journal=Exp. Math. | volume=18 | number=2 | pages=213–222 | year=2009 | issn=1058-6458 | mr=2549691 }}
10. ^{{cite journal | zbl=1226.11119 | last1=Fukuda | first1=Takashi | last2=Komatsu | first2=Keiichi | title=Weber's class number problem in the cyclotomic -extension of III | journal=Int. J. Number Theory | volume=7 | number=6 | pages=1627–1635 | year=2011 | issn=1793-7310 | mr=2835816 | url=http://www.worldscientific.com/doi/abs/10.1142/S1793042111004782 | doi=10.1142/S1793042111004782}}
11. ^{{cite journal | zbl=1205.11116 | last=Morisawa | first=Takayuki | title=A class number problem in the cyclotomic -extension of | journal=Tokyo J. Math. | volume=32 | number=2 | pages=549–558 | year=2009 | issn=0387-3870 | mr=2589962 | url=http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=morisawa&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=2589962 | doi=10.3836/tjm/1264170249}}
12. ^{{Citation | last=Stark| first=Harold| author-link=Harold Stark| title=Some effective cases of the Brauer–Siegel theorem| journal=Inventiones Mathematicae| year = 1974| volume=23| pages=135–152| doi=10.1007/bf01405166}}
13. ^{{Citation | last=Odlyzko| first=Andrew| author-link=Andrew Odlyzko| title=Some analytic estimates of class numbers and discriminants| journal=Inventiones Mathematicae| year = 1975| volume= 29| number=3| pages=275–286| doi=10.1007/bf01389854}}
14. ^{{Citation | last=Murty| first=V. Kumar| author-link=V. Kumar Murty| title=Class numbers of CM-fields with solvable normal closure| journal=Compositio Mathematica| year = 2001| volume= 127| number=3| pages=273–287}}
15. ^{{Citation | last=Yamamura| first=Ken| title=The determination of the imaginary abelian number fields with class number one| journal=Mathematics of Computation| year=1994| volume=62| number=206| pages=899–921| doi=10.2307/2153549}}
16. ^{{Citation | last=Louboutin| first=Stéphane| last2=Okazaki| first2=Ryotaro| title=Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one| journal=Acta Arithmetica| year=1994| volume=67| number=1| pages=47–62}}

References

  • {{Neukirch ANT}}

2 : Algebraic number theory|Field theory
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