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.

Nos leemos!

Angel "Java" Lopez
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