# 21 de Marzo, 2020

### 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. Ver Irreducible and prime elementshttps://math.stackexchange.com/questions/1076517/irreducible-and-prime-elements Luego, si quieren algo más en profundidad, y cómo afecta esto a varias estructuras algebraicas: Irreducible Elementshttps://en.wikipedia.org/wiki/Irreducible_element Any Prime is Irreduciblehttps://math.stackexchange.com/questions/69504/any-prime-is-irreducible Prime implies Irreduciblehttps://math.stackexchange.com/questions/1149078/prime-implies-irreducible Irreducible Elements in a Principal Ideal Domain are Primehttps://math.stackexchange.com/questions/770731/irreducible-elements-in-a-pid-are-prime Irreducible Elements in an Unique Factorization Domain are Primehttps://math.stackexchange.com/questions/257955/irreducibles-are-prime-in-a-ufd A principal ideal ring that is not a euclidean ringhttp://www.math.buffalo.edu/~dhemmer/619F11/WilsonPaper.pdf Ring of integers is a Principal Ideal Domain but not a Euclidean domainhttps://math.stackexchange.com/questions/857971/ring-of-integers-is-a-pid-but-not-a-euclidean-domain An example of a principal ideal domain which is not a Euclidean domainhttp://www.maths.qmul.ac.uk/~raw/MTH5100/PIDnotED.pdf En este blog, algo traté del tema cuando comenté Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (4)http://ajlopez.zoomblog.com/archivo/2016/06/28/libro-Abstract-Algebra-de-Carstensen-F.html Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (3)http://ajlopez.zoomblog.com/archivo/2016/06/27/libro-Abstract-Algebra-de-Carstensen-F.html Libro: Abstract Algebra, de Carstensen, Fine, Rosenberg (2)http://ajlopez.zoomblog.com/archivo/2016/06/26/libro-Abstract-Algebra-de-Carstensen-F.html Libro: Abstract Algebra Structure and Application, de Finston y Morandihttp://ajlopez.zoomblog.com/archivo/2016/06/14/libro-Abstract-Algebra-Structure-and-A.html 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. Nos leemos! Angel "Java" Lopezhttp://www.ajlopez.comhttps://twitter.com/ajlopez