Ideal number

Last updated November 01, 2024

In number theory 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 Ernst Kummer, and led to Richard 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 principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field. This means that there is an element of the ring of integers of the Hilbert class field, which is an ideal number, such that the original nonprincipal ideal is equal to the collection of all multiples of this ideal number by elements of this ring of integers that lie in the original field's ring of integers.

Example

For instance, let $y$ be a root of $y^{2}+y+6=0$ , then the ring of integers of the field $\mathbb {Q} (y)$ is $\mathbb {Z} [y]$ , which means all $a+b\cdot y$ with $a$ and $b$ integers form the ring of integers. An example of a nonprincipal ideal in this ring is the set of all $2a+y\cdot b$ where $a$ and $b$ are 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 $w^{3}-w-1=0$ to $\mathbb {Q} (y)$ , giving $\mathbb {Q} (y,w)$ . An ideal number for the nonprincipal ideal $2a+y\cdot b$ is $\iota =(-8-16y-18w+12w^{2}+10yw+yw^{2})/23$ . Since this satisfies the equation $\iota ^{6}-2\iota ^{5}+13\iota ^{4}-15\iota ^{3}+16\iota ^{2}+28\iota +8=0$ it is an algebraic integer.

All elements of the ring of integers of the class field which when multiplied by $\iota$ give a result in $\mathbb {Z} [y]$ are of the form $a\cdot \alpha +y\cdot \beta$ , where

\alpha =(-7+9y-33w-24w^{2}+3yw-2yw^{2})/23

and

\beta =(-27-8y-9w+6w^{2}-18yw-11yw^{2})/23.

The coefficients α and β are also algebraic integers, satisfying

\alpha ^{6}+7\alpha ^{5}+8\alpha ^{4}-15\alpha ^{3}+26\alpha ^{2}-8\alpha +8=0

and

\beta ^{6}+4\beta ^{5}+35\beta ^{4}+112\beta ^{3}+162\beta ^{2}+108\beta +27=0

respectively. Multiplying $a\cdot \alpha +b\cdot \beta$ by the ideal number $\iota$ gives $2a+b\cdot y$ , 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.

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 Lejeune 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" (Edwards 1977, p. 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 $p\equiv 1{\pmod {\lambda }}$ into "complex numbers composed of $\lambda$ th 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 some skepticism.

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.

Related Research Articles

<span class="mw-page-title-main">Fundamental theorem of arithmetic</span> Integers have unique prime factorizations

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 order of the factors. For example,

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal. Some authors such as Bourbaki refer to PIDs as principal rings.

<span class="mw-page-title-main">Factorization</span> (Mathematical) decomposition into a product

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, $3 \times 5$ is an integer factorization of $15$ , and $(x - 2)(x + 2)$ is a polynomial factorization of $x 2 - 4$ .

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

In mathematics, the ideal class group of an algebraic number field $K$ is the quotient group $J K / P K$ where $J K$ is the group of fractional ideals of the ring of integers of $K$ , and $P K$ is its subgroup of principal ideals. The class group is a measure of the extent to which unique factorization fails in the ring of integers of $K$ . The order of the group, which is finite, is called the class number of $K$ .

<span class="mw-page-title-main">Algebraic number theory</span> Branch of number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials $with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial splits into linear terms when reduced mod . That is, it determines for which prime numbers the relation$

In mathematics, the adele ring of a global field is a central object of class field theory, a branch of algebraic number theory. It is the restricted product of all the completions of the global field and is an example of a self-dual topological ring.

In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:

In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

In mathematics, the ring of integers of an algebraic number field $is the ring of all algebraic integers contained in . An algebraic integer is a root of a monic polynomial with integer coefficients: . This ring is often denoted by or . Since any integer belongs to and is an integral element of, the ring is always a subring of .$

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.

In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.

Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x³ ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x³ ≡ p is solvable if and only if x³ ≡ q is solvable.

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy.

In mathematics, an algebraic number field is an extension field $of the field of rational numbers such that the field extension has finite degree . Thus is a field that contains and has finite dimension when considered as a vector space over .$

In algebraic number theory the n-th power residue symbol is a generalization of the (quadratic) Legendre symbol to n-th powers. These symbols are used in the statement and proof of cubic, quartic, Eisenstein, and related higher reciprocity laws.

In algebraic number theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher powers. It is one of the earliest and simplest of the higher reciprocity laws, and is a consequence of several later and stronger reciprocity laws such as the Artin reciprocity law. It was introduced by Eisenstein, though Jacobi had previously announced a similar result for the special cases of 5th, 8th and 12th powers in 1839.

This is a glossary of concepts and results in number theory, a field of mathematics. Concepts and results in arithmetic geometry and diophantine geometry can be found in Glossary of arithmetic and diophantine 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.

External links

Ideal Numbers, Proof that the theory of ideal numbers saves unique factorization for cyclotomic integers at Fermat's Last Theorem Blog.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.