You're right, but phrasing things in terms of addition and zero is possibly taking some liberties with the definition of group. Even if additive groups and general groups are isomorphic, I agree it's a bit misleading to use additive group naming to provide a lay definition of "group".
That's a statement that goes quite against the claims of the article and I kinda want to ask for a citation ... But I already feel that this discussion is way over my head and I wouldn't understand any journal articles that would clear this up ...
What they are called might not be the point, but calling them prime in a context in which prime has a precise meaning while they are not can be confusing. Especially since the difference between prime and irreducible is subtle and it's the whole reason we have non unique factorization domains, which is the main topic of the article
It's mostly just pointing out that in a particular extension of the integers, if we attach the square root of -5, then (1 + sqrt-5) * (1 - sqrt-5) = 2 * 3.
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).
> Mathematicians call this new system a number “ring”; they can create an infinite variety of them, depending on the new values they choose to incorporate.
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.
I’m not sure if this is what you are missing, but: Rings are closed, so you have to add an infinity of new numbers, not just the literal sqrt(5). a + b*sqrt(5) is a convenient way to denote that all those numbers are in your system. So in some sense, once you’ve decided to incorporate sqrt(5), it really is only a ring once you add that part.
Obviously there are tons of rings that have nothing to do with this recipe... maybe that is your nitpick?
"Number ring" has a specific meaning [0]. I am not sure if it is equivalent to the integers adjoined with some element though.
>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.
It's a ring, but not all rings have to be in that form. I can see that you think they weakly imply the latter, which really is just a bug in our (humans') natural language processing skills. :)
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[ 24.2 ms ] story [ 248 ms ] threadThose 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.
Why would you say that?
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.
[0] http://mathworld.wolfram.com/NumberRing.html
(inb4 all the Proust fans downmod me to oblivion)