In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekind's definition of ideals for rings. An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the Principalization theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means there is an element of the ring of integers of the class field, which is an ideal number, such that all multiples times elements of this ring of integers lying in the ring of integers of the original field define the nonprincipal ideal. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Julius Wilhelm Richard Dedekind (October 6, 1831 - February 12, 1916) was a German mathematician and Ernst Eduard Kummers closest follower in arithmetic. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... A ring is usually anything resembling a circle, or a noise that cycles rapidly. ... Class field theory is a branch of algebraic number theory, including most of the major results that were proved in the period about 1900-1950. ... In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree...

Example

For instance, let y be a root of y² + y + 6 = 0, then the ring of integers of the field is , which means all a + by with a and b integers form the ring of integers. An example of a nonprincipal ideal in this ring is 2a + yb with a and b integers; the cube of this ideal is principal, and in fact the class group is cyclic of order three. The corresponding class field is obtained by adjoining an element w satisfying w³ - w - 1 = 0 to , giving . An ideal number for the nonprincipal ideal 2a + yb is ι = ( − 8 − 16y − 18w + 12w² + 10yw + yw²) / 23. Since this satisfies the equation ι⁶ − 2ι⁵ + 13ι⁴ − 15ι³ + 16ι² + 28ι + 8 = 0 it is an algebraic integer. In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ...

All elements of the ring of integers of the class field which when multiplied by ι give a result in are of the form aα+bβ, where α = ( − 7 + 9y − 33w − 24w² + 3yw − 2yw²) / 23 and β = ( − 27 − 8y − 9w + 6w² − 18yw − 11yw²) / 23. The coefficients α and β are also algebraic integers, satisfying α⁶ + 7α⁵ + 8α⁴ − 15α³ + 26α² − 8α + 8 = 0 and β⁶ + 4β⁵ + 35β⁴ + 112β³ + 162β² + 108β + 27 = 0 respectively. Multiplying aα + bβ by the ideal number ι gives 2a + by, which is the nonprincipal ideal.

History

Kummer first published the failure of unique factorization in cyclotomic fields in 1844 in an obscure journal; it was reprinted in 1847 in Liouville's journal. In subsequent papers in 1846 and 1847 he published his main theorem, the unique factorization into (actual and ideal) primes. In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... 1844 was a leap year starting on Monday (see link for calendar). ... 1847 was a common year starting on Friday (see link for calendar). ... Joseph Liouville (born March 24, 1809, died September 8, 1882) was a French mathematician. ... 1846 was a common year starting on Thursday (see link for calendar). ... 1847 was a common year starting on Friday (see link for calendar). ...

It is widely believed that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Dirichlet told him his argument relied on unique factorization; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. Harold Edwards says the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken" (op cit pp. 79). Kummer's use of the letter λ to represent a prime number, α to denote a λth root of unity, and his study of the factorization of prime number into "complex numbers composed of λth roots of unity" all derive directly from a paper of Jacobi which is concerned with higher reciprocity laws. Kummer's 1844 memoir was in honor of the jubilee celebration of the University of Königsberg and was meant as a tribute to Jacobi. Although Kummer had studied Fermat's Last Theorem in the 1830s and was probably aware that his theory would have implications for its study, it is more likely that the subject of Jacobi's (and Gauss's) interest, higher reciprocity laws, held more importance for him. Kummer referred to his own partial proof of Fermat's Last Theorem for regular primes as "a curiosity of number theory rather than a major item" and to the higher reciprocity law (which he stated as a conjecture) as "the principal subject and the pinnacle of contemporary number theory." On the other hand, this latter pronouncement was made when Kummer was still excited about the success of his work on reciprocity and when his work on Fermat's Last Theorem was running out of steam, so it may perhaps be taken with a grain of salt. Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. ... Gabriel Lamé (July 22, 1795, Tours, France - May 1, 1870, Paris, France) was a French mathematician. ... Peter Gustav Lejeune Dirichlet. ... Kurt Hensel (1861-1941) was a German mathematician, a follower of Leopold Kronecker. ... 1910 in topic: Arts Architecture- Art- Film- Literature- Music- Television Science and technology Aviation- Rail transport- Radio- Science Other topics Australia- Canada- Ireland- South Africa- Sport Births- Deaths Lists of leaders: State leaders - Religious leaders 1910 was a common year starting on Saturday (see link for calendar). ... Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ... In mathematics, reciprocity is applied to a number of theorems, and at times certain relationships. ... 1844 was a leap year starting on Monday (see link for calendar). ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In mathematics, regular primes are a certain kind of prime numbers. ...

The extension of Kummer's ideas to the general case was accomplished independently by Kronecker and Dedekind during the next forty years. A direct generalization encountered formidable difficulties, and it eventually led Dedekind to the creation of the theory of modules and ideals. Kronecker dealt with the difficulties by developing a theory of forms (a generalization of quadratic forms) and a theory of divisors. Dedekind's contribution would become the basis of ring theory and abstract algebra, while Kronecker's would become major tools in algebraic geometry. In abstract algebra, a module is a generalization of a vector space. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In algebraic geometry, divisors are a generalization of subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

References

• Nicolas Bourbaki, Elements of the History of Mathematics. Springer-Verlag, NY, 1999.
• Harold M. Edwards, Fermat's Last Theorem. A genetic introduction to number theory. Graduate Texts in Mathematics vol. 50, Springer-Verlag, NY, 1977.
• C.G. Jacobi, Über die complexen Primzahlen, welche in der theori der Reste der 5ten, 8ten, und 12ten Potenzen zu betrachten sind, Monatsber. der. Akad. Wiss. Berlin (1839) 89-91.
• E.E. Kummer, De numeris complexis, qui radicibus unitatis et numeris integris realibus constant, Gratulationschrift der Univ. Breslau zur Jubelfeier der Univ. Königsberg, 1844; reprinted in Jour. de Math. 12 (1847) 185-212.
• E.E. Kummer, Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren, Jour. für Math. (Crelle) 35 (1847) 327-367.
• John Stillwell, introduction to Theory of Algebraic Integers by Richard Dedekind. Cambridge Mathematical Library, Cambridge University Press, Great Britain, 1996.