Articles

Gaussian Integers and the Origins of Algebraic Number Theory II


Mathematics is the queen of science, and Number Theory is the queen of mathematics.
C. F. Gauss

Algebraic number theory was created in the second half of the 19th century in the works of mathematicians Ernest Kummer (1810-1893), Richard Dedekind (1831-1916) and Leopold Kronecker (1823-1891). This theory had its origins when the German mathematician Carl F. Gauss (1777-1855) extended the idea of ​​integer by defining the ring of Gaussian algebraic integers, Zi, and later in an attempt to demonstrate Fermat's Last Theorem. Algebraic number theory is one of the most beautiful and profound theories in all mathematics.

The first motivation of this investigation concerns the generalization of the single representation theorem of integers as a product of prime numbers, less than the order of factors, to algebraic integers. Gauss introduced the ring of algebraic integers, Zi, during his investigation of bikadratic residues, and showed that in this ring factorization in prime elements exists, and is unique less than the order of factors.

The factorization of a number depends very much on the ring to which it belongs, and therefore, in order to generalize the uniqueness of integer factorization, it is necessary to work on appropriate sub-rings of the body of complex numbers.

The second motivation for the study of algebraic number arithmetic stems from the Theory of Diophantine Equations. For example, a quadratic shape defined over a ring A is a homogeneous polynomial such that the coefficients are elements of A, that is, polynomials of the shape f(x, y) = Ax2 + Bxy + Cy2 Where THE, B, Ç belong to ring A. If we take the quadratic form over the ring of integers

f(x, y) = x2 - D y2

Where D is an integer and ÖD is not an integer, it can be written in the form

f(x, y) = x2 - D y2 = (x - ÖD y). (x + ÖD y).

Therefore, the question about the possibility of integer representation r per r = The2 - DB2 = f(The, B) Where The and B are integers, is reformulated as a matter of factoring algebraic numbers of the ring ZÖD, that is, numbers of the form The + BÖD.

These motivations make clear the importance of Z Ö ringsD and Z i.

In the early 1840s, Kummer considered the shape ring of numbers.

<>

TheP-<>

1V<>

P-<>

1<>

+ TheP-2 V<>

P-<>

2<>

+… + The1V<>

+ The0

<>

Where TheP-1, TheP-2,… , The1 and The0 are whole numbers, P is an odd prime number and V a primitive root P-th of the unit, that is, a complex number V such that VP = 1 and V Como 1. Since this ring generally does not have the property of single factorization in prime numbers, Kummer fixed this by introducing the notion of “ideal numbers”, which gave rise to the notion of “ideal” due to Dedekind, and showed that was worth the unique factorization into ideal prime numbers. With this concept, he demonstrated Fermat's Last Theorem, in many cases new at the time, using the identity

xP - yP = (x - y) (x - Vy)… (x - VP - 1y ) .

This theory took a different form from what Kummer bequeathed us. However, Kummer's profound results on cyclotomic bodies, ie bodies of the form Q (w) where w is a primitive root no-th of the unit, served as a paradigm for later researchers.

It took about 30 years for Kronecker and Dedekind to find the correct generalization of the ideal numbers. It was noted that it was necessary to define the notion of algebraic integer.

An algebraic integer is a particular type of complex number, that is, a complex number that is the solution of a polynomial equation.

<>

Thenoxno<>

+ Theno-1 xno-1+… + The1x + The0 = 0,

where all the coefficients Theno, Theno-1,… , The1, The0 are integers. For example, the imaginary unit, i, is an algebraic integer because it satisfies the equation x2 + 1 = 0. The square root of 7, Ö7, is an algebraic integer because it satisfies the equation x2 - 7 = 0. Note that the numbers i, Ö7 are examples of algebraic integers and are not integers.

Algebraic integer rings represent the central concept of Algebraic Number Theory. To be exact: one body of algebraic numbers, K, and its corresponding algebraic integer ring, DK. A body of algebraic numbers, K, is a subbody of the body of complex numbers that, when viewed as a vector space over rationals, Q has a finite dimension. The algebraic integers contained in K form a ring DK, which is the appropriate structure for the generalization of single factorization in prime numbers.

Generally speaking: if w is an arbitrary algebraic number and we take the body K = Q (w) then consider the distinguished sub-ring DK of K called the ring of the algebraic integers of K. The elements of DK are complex numbers contained in K = Q (w) that are solutions of polynomial equations

<>

Thenoxno<>

+ Theno-1 xno-1+… + The1x + The0 = 0,

where all the coefficients Theno, Theno-1,… , The1, The0 are integers.

Note that the relationship between DK and K is analogous to the relationship between Z and Q. However, prime factorization tends to fail for integer ring elements, but not for ideals.

We draw the reader's attention to the fact that when we take the body K = Q (w), where w is an arbitrary algebraic number, then the ring of algebraic integers is not always of the form DK = Z w. On the other hand, it is true that Z w is contained in DK,because DK is a ring containing w. For example, Q (Ö5) is a body of algebraic numbers. In fact, the complex number Ö5 is the root of the polynomial. P(x) = x2 - 5, therefore an algebraic number, and Q (Ö5) is a vector space of finite dimension equal to 2 over Q, a base being the set {1, Ö5}. However, Z Ö5 is not your integer ring. In fact, the complex number (1 + Ö5) / 2 is the root of the polynomial. P(x) = x2 - x - 1, therefore an algebraic integer belonging to Q (Ö5). Thus, the complex number (1 + Ö5) / 2 belongs to the ring of algebraic integers DK, but does not belong to Z (Ö5) because the number 1/2 is not an integer.

The mathematician Dedekind reformulated Kummer's concept of the ideal number, proposing the fundamental key concept of the "ideal" that remains today. Dedekind's definition is distinct from Kummer's definition, but it is shown that they are equivalent. In this theory, the essential building blocks are prime ideals. It is demonstrated that in algebraic integer rings every nonzero ideal has unique factorization in powers of prime ideals.

The theory of algebraic integer ring ideals was created to provide new classical problem solving methods of Number Theory. The development of methods in Algebraic Number Theory remains an important area of ​​research in Number Theory.

The abstraction of the most essential properties of algebraic integer rings gave rise to axioms that defined a new class of rings called Dedekind Domains, as demonstrated by the brilliant German mathematician Emmy Noether (1882-1935). Dedekind's domain class is much larger than the original class of algebraic integer rings. The basic invariant of a Dedekind ring is its group of ideal classes, class group in English, and its cardinality is called the number of ideal classes, class number in English. This is usually an infinite abelian group. However, it is always a finite group for algebraic integer rings.

If we consider a body of algebraic numbers K and its ring of algebraic integers DK, it is shown that the algebraic integer ring, DK, is a domain of Dedekind. Being DK a ring of algebraic integers the class group is finite and it is shown that the class number equals 1 if, and only if, the integer ring, DK, has the property of unique factorization.

The research of the arithmetic properties of the integer ring of a body of algebraic numbers is one of the main objects of investigation in Algebraic Number Theory. There are three methods for investigating D arithmetic.K. Kronecker considered polynomials with D coefficientsK. Dedekind introduced the notion of ideals into DKby defining one of the most important concepts in algebra. Hensel introduced the method that is currently called localization.

Much of classical Number Theory can be expressed in the context of Algebraic Number Theory, and this theory has moved from tool to object of essential investigation in Number Theory. This view was greatly emphasized by the German mathematician David Hilbert (1862-1943) who had a huge influence on the development of Number Theory. As a result, Algebraic Number Theory is a fertile, prosperous, and important branch of mathematics, with profound methods and applications not only in Number Theory itself, but also in Group Theory, Algebraic Geometry, Commutative Algebra, Topology. , Analysis and K-Theory.

Back to columns

<