New Number Systems Seek Their Lost Primes
quantamagazine.org
quantamagazine.org
Those factors are irreducibles, but they aren't primes, which is the reason why the uniqueness of factorization fails.
A nonzero non unit element of a ring is called prime if p|ab implies p|a or p|b.
A nonzero non unit element of a ring is called irreducible if p=ab implies that a is a unit or b is a unit (invertible element).
Primes are irreducibles in an integral domain, but the converse is true in unique factorization domains and Z[√-5] is not one.
For a more serious reference any abstract algebra book covering rings and UFDs should do, for example it is in Dummit & Foote.
3 divides the left hand side, but it is not a divisor of either term. This is different than in the usual integers, since there are irreducible terms (not divisble by anything) that aren't prime (if they divide a product, they divide one of the factors).
Isn't this a little wrong?
As far as I remember ring is any set with multiplication, negation, and addition defined so that they satisfy a few conditions. No need for the "+ b * something" part. The usual integer numbers we use form a ring too, as well as booleans.
I might be missing something, and it's irrelevant to the main subject of the article.
The number system that you generate by adding sqrt(5) to the integers is a ring because it satisfies the definition. And you can create an infinite variety of rings by adding new values to the integers.
But yeah, I'm nitpicking.
Obviously there are tons of rings that have nothing to do with this recipe... maybe that is your nitpick?
But yeah, the integers form a ring in their own right. They call the new system a ring, but that doesn't mean the old system wasn't also a ring.
Also not all rings are of that form - as you say a ring is a very general algebraic structure.
>In fact the usual integer numbers we use form a ring too.
Depending on definitions, the integers can be viewed as the ring of even numbers adjoined with the element 1. This does require that we do not define rings to necessarily contain 1. In my experience this definition is is becoming more of a historical footnote though.
There is also a natural generalization of adjoining multiple elements to the natural numbers (which still results in a ring) In this case, the natural numbers would just be a special case of adjoining 0 elements.
The ring of real numbers, in contrast, cannot be constructed by adjoining any finite set of elements to the integers.
The ring of integers mod n is also a ring, but does not contain the integers as a subring (and therefore cannot be thought of as the integers adjoined with any set (finite or infinite) of elements).
The polynomials with coeficients mod n form a non-finite ring which does not contain the integers.
I suspect the point that the author was attempting to make was just that Z[√5] formed a type of structure that mathematicians are familiar with.
(inb4 all the Proust fans downmod me to oblivion)