# topics on ideal class groups and discriminants

## Ideal Class Groups, Class Numbers and Discriminants (http://planetmath.org/browse/objects/11R29/MSC 11R29)

Let $K$ be a number field  (that is, a finite extension  of the rational numbers $\mathbb{Q}$) and let $\mathcal{O}_{K}$ be the ring of integers  in $K$. The ring of integers of $K$ is the analogue of $\mathbb{Z}$ in $\mathbb{Q}$. As we know, $\mathbb{Z}$ enjoys the property that any number can be factored uniquely as a product  of powers of primes. In particular, $\mathbb{Z}$ is a UFD and a PID (principal ideal domain  ). When is $\mathcal{O}_{K}$ a UFD or a PID? This is a very hard question to answer. The and class number of $K$ are objects that measures how far $\mathcal{O}_{K}$ is from actually being a PID. In that sense, the class groups measure the arithmetic  complexity of a number field. We include the basic definition of class group here for convenience of the reader:

###### Definition 1.

The class group, $\operatorname{Cl}(K)$, of a number field $K$ is defined to be the quotient group  of all fractional ideals   of $K$ modulo principal fractional ideals. The size of the class group $|\operatorname{Cl}(K)|$ is called the class number of $K$ and it is usually written $h_{K}$.

### 1.1 Basic Definitions

1. 1.

The definition of class group and class number can be found at the entry ideal class. Notice that the ideal classes form an abelian group  (the entry also discusses properties of ideal classes).

2. 2.

The Hilbert class field  of $K$, usually denoted by $H$, is the maximal unramified abelian extension   of $K$. In particular, the Galois group  $\operatorname{Gal}(H/K)$ is isomorphic   to the class group of $K$ which is the link between ramification, class field theory and class numbers. The entry on the existence of the Hilbert class field (http://planetmath.org/ExistenceOfHilbertClassField) discusses alternative characterizations  of $H$.

3. 3.

### 1.2 Computing Class Groups and Class Numbers

1. 1.
2. 2.
3. 3.

Using Minkowski’s constant to find a class number (contains examples).

### 1.3 Divisibility Properties of Class Numbers

The entry on unramified extensions and class number divisibility is a corollary of the existence of the Hilbert class field and clarifies the connection between the prime divisors  of $h_{K}$ and the unramified abelian extensions of $K$.

1. 1.

Class number divisibility in extensions: $F/K$ Galois, $[F:K]$ not divisible by $p$. Then $p|h_{K}$ implies $p|h_{F}$.

2. 2.

Class number divisibility in cyclic extensions: $F/K$ Galois and cyclic with $[F:K]$ not divisible by $p$ and $p$ does not divide the class number of intermediate extensions. Then if $p|h_{F}$ then $p^{f}|h_{F}$ for some $f$ (see entry for details).

3. 3.

Extensions without unramified subextensions and class number divisibility: $F/K$ such that there are no non-trivial abelian unramified subextensions. Then $h_{K}|h_{F}$.

4. 4.

Class number divisibility in $p$-extensions (http://planetmath.org/ClassNumberDivisibilityInPExtensions): $F/K$ is a Galois $p$-extension which is ramified at most at one prime. If $p|h_{F}$ then $p|h_{K}$.

5. 5.

Push-down theorem on class numbers: $F/K$ is a $p$-extension which is ramified exactly at one prime and this prime is totally ramified. If $p|h_{F}$ then $p|h_{K}$.

### 1.4 Class Numbers of Cyclotomic Fields

Cyclotomic fields  have been the object of extensive study. For example, they are crucial in some of the “easy” cases of Fermat’s Last Theorem. For any number $n$, let $\zeta_{n}$ be a primitive $n$th root of unity  . The field $K=\mathbb{Q}(\zeta_{n})$ is a cyclotomic field. We denote its class number by $h_{n}$.

1. 1.
2. 2.
3. 3.
4. 4.

Thaine’s theorem is the counterpart of Stickelberger’s theorem for totally real fields.

5. 5.

Vandiver’s conjecture states that a prime number $p$ cannot divide the class number of the maximal real subfield  of $\mathbb{Q}(\zeta_{p})$.

6. 6.

The index of the group of cyclotomic units in the full unit groups is exactly the class number of the maximal real subfield of $\mathbb{Q}(\zeta_{p})$.

### 1.5 Discriminants and Related Results

1. 1.

Definition of discriminant (http://planetmath.org/Discriminant) (also discusses the relationship with discriminants in other contexts).

2. 2.

A related concept: the root-discriminant.

3. 3.

Hermite’s theorem on extensions which are unramified outside a fixed set of primes.

## References

1. 1.
2. 2.

Daniel A. Marcus, Number Fields, Springer, New York.

3. 3.

K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1998.

4. 4.

Lawrence C. Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York.

Note: If you would like to contribute to this entry, please send an email to the author (alozano).

 Title topics on ideal class groups and discriminants Canonical name TopicsOnIdealClassGroupsAndDiscriminants Date of creation 2013-03-22 15:07:44 Last modified on 2013-03-22 15:07:44 Owner alozano (2414) Last modified by alozano (2414) Numerical id 13 Author alozano (2414) Entry type Topic Classification msc 11R29 Related topic IdealClass Related topic BibliographyForNumberTheory Related topic ClassNumberDivisibilityInCyclicExtensions Related topic ClassNumberDivisibilityInPExtensions Related topic ClassNumberFormula Related topic UnramifiedExtensionsAndClassNumberDivisibility Related topic PushDownTheoremOnClassNumbers Related topic ClassNumberDivisibilityInExtension Defines ideal class group