Quadratic integer

In number theory, quadratic integers are a generalization of the rational integers to quadratic fields. Important examples include the Gaussian integers and the Eisenstein integers. Though they have been studied for more than a hundred years, many open problems remain.

1 Definition
2 Examples
3 Class number
4 See also
5 Notes
6 References

Definition

Quadratic integers are solutions of equations of the form:

x² + Bx + C = 0

for integers B and C. Such solutions have the form a + ωb, where a, b are integers, and where ω is defined by;

$\omega = \begin{cases} \sqrt{D} & \mbox{if }D \equiv 2, 3 \pmod{4} \\ {{1 %2B \sqrt{D}} \over 2} & \mbox{if }D \equiv 1 \pmod{4} \end{cases}$

(D is a square-free integer).

This characterization was first given by Richard Dedekind in 1871.^[1]^[2] Fixing a square-free integer D, the quadratic integer ring Z[ω] = {a + ωb : a, b ∈ Z} is a subring of the quadratic field $\mathbf{Q}(\sqrt{D})$ . Moreover, Z[ω] is the integral closure of Z in $\mathbf{Q}(\sqrt{D})$ . In other words, it is the ring of integers $\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ of $\mathbf{Q}(\sqrt{D})$ and thus a Dedekind domain.

Examples

A classic example is $\mathbf{Z}[\sqrt{-1}]$ , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.^[3]
The elements in $\mathcal{O}_{\mathbf{Q}(\sqrt{-3})} = \mathbf{Z}\left[{{1 %2B \sqrt{-3}} \over 2}\right]$ are called Eisenstein integers.
In contrast, $\mathbf{Z}[\sqrt{-3}]$ is not even a Dedekind domain.

Class number

Equipped with the norm

$N(a %2B b\sqrt{D}) = a^2 - Db^2$ ,

$\mathcal{O}_{\mathbf{Q}(\sqrt{D})}$ is an Euclidean domain (and thus a unique factorization domain) when $D = -1, -2, -3, -7, -11.$ ^[4] On the other hand, it turned out that $\mathbf{Z}[\sqrt{-5}]$ is not a UFD because, for example, 6 has two distinct factorizations into irreducibles:

$6 = 2(3) = (1 %2B \sqrt{-5}) (1 - \sqrt{-5}).$

(In fact, $\mathbf{Z}[\sqrt{-5}]$ has class number 2.^[5]) The failure of the unique factorization led Ernst Kummer and Dedekind to develop a theory that would enlarge the set of "prime numbers"; the result was the notion of ideals and the decomposition of ideals by prime ideals (cf. splitting of prime ideals in Galois extensions)

Being a Dedekind domain, a quadratic integer ring is a UFD if and only if it is a principal ideal domain (i.e., its class number is one.) However, there are quadratic integer rings that are principal ideal domains that are not Euclidean domains. For example, $\mathbf{Q}[\sqrt{-19}]$ has class number 1 but its ring of integers is not Euclidean.^[5] There are effective methods to compute ideal class groups of quadratic integer rings, but many theoretical questions about their structure are still open after a hundred years.

Notes

^ Dedekind 1871, Supplement X, p. 447
^ Bourbaki 1994, p. 99
^ Dummit, pg. 229
^ Dummit, pg. 272
^ ^a ^b Milne, pg. 64

References

Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, ISBN 978-3-540-64767-6, MR 1290116 . Translated from the original French by John Meldrum
Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg, http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN30976923X&DMDID=dmdlog1 . Retrieved 5. August 2009
Dummit, D. S., and Foote, R. M., 2004. Abstract Algebra, 3rd ed.
J.S. Milne. Algebraic Number Theory, Version 3.01, September 28, 2008. online lecture note