This is an incomplete list of

number fields In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

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

Definition

The class number of a number field is by definition the order of the

ideal class group In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...

of its

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

. Thus, a number field has class number 1 if and only if its ring of integers is a

principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, ...

(and thus a

unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...

). The

fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...

says that Q has class number 1.

Quadratic number fields

These are of the form ''K'' = Q(), for a

square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-f ...

''d''.

Real quadratic fields

''K'' is called real quadratic if ''d'' > 0. ''K'' has class number 1 for the following values of ''d'' : * 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 (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() for ''d'' = 229 and ''d'' = 257 both have class number greater than 1 (in fact equal to 3 in both cases). The density of such primes for which Q() does have class number 1 is conjectured to be nonzero, and in fact close to 76%, however it is not even known whether there are infinitely many real quadratic fields with class number 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. (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 In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...

) 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:Tables available at Pari source code
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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 number In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic ...

(negated), while that for −31 is the reciprocal of the

supergolden ratio In mathematics, two quantities are in the supergolden ratio if the quotient of the larger number divided by the smaller one is equal to :\psi = \frac which is the only real solution to the equation x^3 = x^2+1. It can also be represented using ...

. The polynomials defining the complex cubic fields that have class number one and discriminant greater than −500 are:

Cyclotomic fields

The following is a complete list of ''n'' for which the field Q(ζ_''n'') has class number 1: * 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. On the other hand, the maximal real subfields Q(cos(2π/2^''n'')) of the 2-power cyclotomic fields Q(ζ_2^''n'') (where ''n'' is a positive integer) are known to have class number 1 for n≤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, and later improved this bound to 10⁹. 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. 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.

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 number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyno ...

. In 1974,

Harold Stark Harold Mead Stark (born August 6, 1939 in Los Angeles, California) is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earli ...

conjectured that there are finitely many CM fields of class number 1. He showed that there are finitely many of a fixed degree. Shortly thereafter,

Andrew Odlyzko Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...

showed that there are only finitely many ''Galois'' CM fields of class number 1. 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. 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. Combining this list with the work of Stéphane Louboutin and Ryotaro Okazaki provides a full list of quartic CM fields of class number 1.

Notes

References

*{{Neukirch ANT Algebraic number theory Field (mathematics)