List of number fields with class number one

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.

{{Main|Quadratic number field}}

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

{{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*, ...Chapter I, section 6, p. 37 of {{harvnb|Neukirch|1999}}{{cite journal | zbl=1152.11328 | last=Dembélé | first=Lassina | title=Explicit computations of Hilbert modular forms on $\backslash mathbb\; Q(\backslash sqrt\{5\})$ | 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| s2cid=9088028 }}

(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).H. Cohen, A Course in Computational Algebraic Number Theory, GTM 138, Springer Verlag (1993), Appendix B2, p.507 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%,{{cite book | last1=Cohen | first1=H. | last2=Lenstra | first2=H. W. | title=Number Theory Noordwijkerhout 1983 | chapter=Heuristics on class groups of number fields | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | volume=1068 | date=1984 | isbn=978-3-540-13356-8 | doi=10.1007/bfb0099440 | doi-access=free | url=https://scholarlypublications.universiteitleiden.nl/access/item%3A2721029/view | access-date=2025-04-20 | pages=33–62}}

however it is not even known whether there are infinitely many real quadratic fields with class number 1.

{{Main|Heegner number}}

K has class number 1 exactly for the 9 following negative values of d:

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

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

{{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 polynomials defining the totally real cubic fields that have discriminants less than 500 with class number one are:[http://pari.math.u-bordeaux1.fr/pub/pari/packages/nftables/ Tables available at Pari source code]

{{columns-list|colwidth=35em|

• 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)

}}

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:

{{columns-list|colwidth=35em|

• 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)

}}

{{Main|Cyclotomic field}}

The following is a complete list of thirty n for which the field $\backslash mathbb\{Q\}$(ζ_n) has class number 1:{{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 }}{{Cite OEIS |A005848 |Cyclotomic fields with class number 1 (or with unique factorization). |access-date=2024-03-20 }}

• 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 $\backslash mathbb\{Q\}$(ζ_2n) = $\backslash mathbb\{Q\}$(ζ_n) when n is odd.)

On the other hand, the maximal real subfields $\backslash mathbb\{Q\}$(cos(2π/2ⁿ)) of the 2-power cyclotomic fields $\backslash mathbb\{Q\}$(ζ_2ⁿ) (where n is a positive integer) are known to have class number 1 for n≤8,{{cite journal | last=Miller | first=John C. | title=Class numbers of totally real fields and applications to the Weber class number problem | journal=Acta Arithmetica | volume=164 | issue=4 | date=2014 | issn=0065-1036 | doi=10.4064/aa164-4-4 | doi-access=free | pages=381–397 | url=https://www.impan.pl/shop/publication/transaction/download/product/83125?download.pdf | access-date=2025-04-20}} and

it is conjectured that they have class number 1 for all n. Weber showed{{cite journal | last=Weber | first=H. | title=Theorie der Abel'schen Zahlkörper | journal=Acta Mathematica | volume=8 | date=1886 | issn=0001-5962 | doi=10.1007/BF02417089 | doi-access=free | pages=193–263 | url=https://projecteuclid.org/journals/acta-mathematica/volume-8/issue-none/Theorie-der-Abelschen-Zahlk%c3%b6rper/10.1007/BF02417089.pdf | access-date=2025-04-22}} 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,{{cite journal | zbl=1189.11033 | last1=Fukuda | first1=Takashi | last2=Komatsu | first2=Keiichi | title=Weber's class number problem in the cyclotomic $\backslash mathbb\; Z\_2$-extension of $\backslash mathbb\; Q$ | journal=Exp. Math. | volume=18 | number=2 | pages=213–222 | year=2009 | issn=1058-6458 | mr=2549691 | doi=10.1080/10586458.2009.10128896 | s2cid=31421633 }} and later improved this bound to 10⁹.{{cite journal | zbl=1226.11119 | last1=Fukuda | first1=Takashi | last2=Komatsu | first2=Keiichi | title=Weber's class number problem in the cyclotomic $\backslash mathbb\; Z\_2$-extension of $\backslash mathbb\; Q$ III | journal=Int. J. Number Theory | volume=7 | number=6 | pages=1627–1635 | year=2011 | issn=1793-7310 | mr=2835816 | doi=10.1142/S1793042111004782| s2cid=121397082 }} These fields are the n-th layers of the cyclotomic $\backslash mathbb\{Z\}$₂-extension of $\backslash mathbb\{Q\}$. Also in 2009, Morisawa showed that the class numbers of the layers of the cyclotomic $\backslash mathbb\{Z\}$₃-extension of $\backslash mathbb\{Q\}$ have no prime factor less than 10⁴.{{cite journal | zbl=1205.11116 | last=Morisawa | first=Takayuki | title=A class number problem in the cyclotomic $\backslash mathbb\; Z\_3$-extension of $\backslash mathbb\; Q$ | journal=Tokyo J. Math. | volume=32 | number=2 | pages=549–558 | year=2009 | issn=0387-3870 | mr=2589962 | url=https://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| doi-access=free }} Coates has raised the question of whether, for all primes p, every layer of the cyclotomic $\backslash mathbb\{Z\}$_p-extension of $\backslash mathbb\{Q\}$ has class number 1.{{cite conference |last=Coates |first=John |date=2011 |title=The enigmatic Tate-Shafarevich group |work=Proceedings of the Fifth International Congress of Chinese Mathematicians |location=Beijing, China }}

{{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.{{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

| issue=2

| pages=135–152

| doi=10.1007/bf01405166

| bibcode=1974InMat..23..135S

| hdl=10338.dmlcz/120573

| s2cid=119482000

| hdl-access=free

}} 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.{{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

| bibcode=1975InMat..29..275O

| s2cid=119348804

}}

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.{{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

| doi=10.1023/A:1017589432526

| doi-access=free

}}

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.{{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

| jstor=2153549

| bibcode=1994MaCom..62..899Y

| doi-access=free

}} Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.{{Citation

| last1=Louboutin

| first1=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

| doi=10.4064/aa-67-1-47-62

| doi-access=free

}}

• Class number problem
• Class number formula
• Brauer–Siegel theorem

{{reflist}}

• {{Neukirch ANT}}

Category:Algebraic number theory

Category:Field (mathematics)