Angel "Java" Lopez en Blog


Publicado el 5 de Abril, 2020, 19:51

En estos días estuve leyendo temas de teoría algebraica de números. Algo publiqué por acá en Números Irreducibles y Primos. Hoy leo en el libro Algebraic Number Theory de Romyar Sharifi:

At its core, the ancient subject of number theory is concerned with the arithmetic of the integers. The Fundamental Theorem of Arithmetic, which states that every positive integer factors uniquely into a product of prime numbers, was contained in Euclid’s Elements, as was the infinitude of the set of prime numbers. Over the centuries, number theory grew immensely as a subject, and techniques were developed for approaching number-theoretic problems of a various natures. For instance, unique factorization may be viewed as a ring-theoretic property of Z, while Euler used analysis in his own proof that the set of primes is infinite, exhibiting the divergence of the infinite sum of the reciprocals of all primes.

Es importante conocer que la factorización única NO SIEMPRE está presente en otros anillos. Es parte de lo que la teoría algebraica de números tiene para ofrecernos.

Algebraic number theory distinguishes itself within number theory by its use of techniques
from abstract algebra to approach problems of a number-theoretic nature. It is also often considered, for this reason, as a subfield of algebra. The overriding concern of algebraic number theory is the study of the finite field extensions of Q, which are known as number fields, and their rings of integers, analogous to Z.

Curiosamente las extensions de campos comenzaron a aparecer en los trabajos para resolver los ecuaciones de grado mayor que 2. Pero si en esas extensions, definimos algo como "enteros", se nos abre la puerta a estudiar nuevos sistemas de números.

The ring of integers O of a number field F is the subring of F consisting of all roots of all
monic polynomials in Z[x]. Unlike Z, not all integer rings are UFDs, as one sees for instance
by considering the factorization of 6 in the ring Z[√−5]. However, they are what are known
as Dedekind domains, which have the particularly nice property that every nonzero ideal factors uniquely as a product of nonzero prime ideals, which are all in fact maximal. In essence, prime ideals play the role in O that prime numbers do in Z.

UFD es Unique Factorization Domain, dominio con factorización única. El caso del 6 mencionado se refiere a que 6 = 3 * 2 pero también en ese anillo Z[√−5]  el 6 es igual a (1 + √−5)(1 - √−5) y esos dos pares de factores no se dividen entre sí. Es un resultado un tanto inesperado, pero sumamente interesante.

Para recuperar la factorización única, hay que reemplazar los enteros por ideales, conjuntos de elementos de un anillo.

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Angel "Java" Lopez

Publicado el 4 de Abril, 2020, 17:42

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Angel "Java" Lopez

Publicado el 31 de Marzo, 2020, 11:04

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Angel "Java" Lopez

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Angel "Java" Lopez

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Angel "Java" Lopez

Publicado el 21 de Marzo, 2020, 13:02

Hoy leo en la introducción del excelente "Algebraic number theory and Fermat's last theorem", de Ian Stewart y David Tall

For organizational reasons rather than mathematical necessity, the book is divided into four parts. Part I develops the basic theory from an algebraic standpoint, introducing the ring of integers of a number field and exploring factorization within it. Quadratic and cyclotomic fields are investigated in more detail, and the Euclidean imaginary fields are classified. We then consider the notion of factorization and see how the notion of a 'prime' p can be pulled apart into two distinct ideas. The first is the concept of being 'irreducible' in the sense that p has no factors other than 1 and p. The second is what we now call 'prime': that if p is a factor of the product ab (possibly multiplied by units—invertible elements) then it must be a factor of either a or b. In this sense, a prime must be irreducible, but an irreducible need not be prime. It turns out that factorization into irreducibles is not always unique in a number field, but useful sufficient conditions for uniqueness may be found. The factorization theory of ideals in a ring of algebraic integers is more satisfactory, in that every ideal is a unique product of prime ideals. The extent to which factorization is not unique can be 'measured' by the group of ideal classes (fractional ideals modulo principal ones).

Es un tema más que interesante: uno, basado en el manejo de enteros y naturales, tiende a poner como equivalentes los conceptos de número primo y número irreducible. Pero se vió (justamente en el siglo XIX, tratando de demostrar el ultimo teorema de Fermat) que no es el caso: hay sistemas de números (anillos) donde no se cumple la equivalencia.


Irreducible and prime elements

Luego, si quieren algo más en profundidad, y cómo afecta esto a varias estructuras algebraicas:

Irreducible Elements

Any Prime is Irreducible

Prime implies Irreducible

Irreducible Elements in a Principal Ideal Domain are Prime

Irreducible Elements in an Unique Factorization Domain are Prime

A principal ideal ring that is not a euclidean ring

Ring of integers is a Principal Ideal Domain but not a Euclidean domain

An example of a principal ideal domain which is not a Euclidean domain

En este blog, algo traté del tema cuando comenté

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (4)

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (3)

Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (2)

Libro: Abstract Algebra Structure and Application, de Finston y Morandi

En esos libros aparece más detallado la evolución del concepto, en especial, la aparición de ideales primos, que de nuevo, tuvo su origen en los intentos de demostración del ultimo teorema de Fermat, por parte de Kummer y sus números ideales, una extension para conseguir la factorización única, luego levantada por Dedekind para formar los ideales primos. Esa extension del concepto de número resultó fructífera, como se ve en los capítulos de los libros mencionados arriba.

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Angel "Java" Lopez

Publicado el 23 de Febrero, 2020, 8:42

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Angel "Java" Lopez

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Angel "Java" Lopez

Publicado el 15 de Febrero, 2020, 14:33

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Angel "Java" Lopez

Publicado el 9 de Febrero, 2020, 11:32

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Angel "Java" Lopez

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Angel "Java" Lopez

Publicado el 25 de Enero, 2020, 11:29

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Angel "Java" Lopez

Publicado el 19 de Enero, 2020, 11:26

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Angel "Java" Lopez

Publicado el 18 de Enero, 2020, 15:42

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Angel "Java" Lopez

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Angel "Java" Lopez

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Angel "Java" Lopez

Publicado el 5 de Enero, 2020, 11:41

En su libro The Principles of Mathematics, Bertrand Russel describe lo que considera matemáticas puras:

Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

Es una definición seca, llevada a la base de toda la matemática pura, pero que deja afuera (o al menos lejos) a cantidad de ramas activas y fructíferas de las matemáticas, puras o no. Pero en aquellos años (1903 fue publicado por primera vez ese libro), Russel buscaba los fundamentos de la matemática, que habían entrado en cierta crisis (recordemos la crítica de Russel al libro sobre conjuntos y lógica de Frege). En su infancia, había encontrado en las matemáticas una roca firme, donde la verdad era clara, en medio quizás de un mundo humano cambiante. Pero para fines del siglo XIX, los fundamentos de las matemáticas no estaban claros ni firmes.

Desde entonces, se ha avanzado, y hoy, gran parte de los matemáticos no se preocupan de los fundamentos, sino del gran juego que es el pensamiento matemático. No lo hacen en general por ser arriesgados, mas bien los fundamentos de la matemática se han ido desarrollando en el siglo XX en varias ramas, desde una teoría de conjuntos más elaborada, hasta teoría de categorías, pasando por el formalismo de Hilbert,  los teoremas de Gödel y la evolución de la lógica matemáticas más allá de verdadero o falso. No es un tema que ha quedado abandonado, sino que sigue siendo bien atendido. Pero hay tantas ramas interesantes y activas, que muchos matemáticos simplemente descansan sobre los hombros de los colegas que se ocupan de los fundamentos.

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Angel "Java" Lopez

Publicado el 4 de Enero, 2020, 15:27

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Angel "Java" Lopez

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Angel "Java" Lopez

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