Quadratic integer

In mathematics, especially 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.

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;

(D is a square-free integer).

This characterization was first given by Dedekind in 1871.^[1][2] Quadratic integers form a subring of a quadratic field , called the quadratic integer ring and denoted by Z[ω]. Moreover, Z[ω] is the integral closure of Z in . In other words, it is the ring of integers of and thus a Dedekind domain.

Examples

• A classic example is , the Gaussian integers, which was introduced by Carl Gauss around 1800 to state his biquadratic reciprocity law.^[3]
• The elements in are called Eisenstein integers.
• In contrast, is not even a Dedekind domain.^{[citation needed]}

Class number

Equipped with the norm

,

is an Euclidean domain (a fortiori, UFD) when D = − 1, − 2, − 3, − 7, − 11.^[4] On the other hand, it turned out that is not a UFD because it contains an irreducible element that is not a prime element. For example, 6 has two distinct factorizations into irreducibles:

(In fact, has class number 2.^[5]) The failure of the unique factorization led 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.

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, 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.

See also

• Fermat's Last Theorem
• Pell's equation

Notes

References

• Bourbaki, Nicolas (1994), Elements of the history of mathematics, Berlin: Springer-Verlag, MR1290116, ISBN 978-3-540-64767-6 . Translated from the original French by John Meldrum
• Dedekind, Richard (1871), Vorlesungen über Zahlentheorie von P.G. Lejeune Dirichlet (2 ed.), Vieweg . 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