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The 4n+3 primes come as they are. 4n+1 primes from the integers can be broken back to smaller prime factors. Even 2 is not prime in that world because it can be factored as (1+i)(1-i).
Does that mean there is no one-to-one correspondence between primes in the integers and primes in the Gaussian integers? 5 is made of two factors, for instance. The density of Gaussian primes must be slightly higher because of that.
I think one could put the Gaussian primes into a one-to-one correspondence with the integers although I don't have a function that would do that at the moment. But that just counts them. There may be some other "measures" associated with them outside of quantity.
Here is some suggestion how to proceed constructing a one-to-one correspondence. First use the one-to-one correspondence between the integers and the rational numbers. Then map the Gaussian integer (or just prime), represented as a+bi, to the rational number represented as a/b. Those two one-to-one correspondences may be able to be composed in some way to get the desired one-to-one correspondence to show the primes and the Gaussian primes have the same countable infinity of elements.
If the normal primes have density 1, the Gaussian primes should have density 2.
What is "density"? One might be able to look at that as the number of primes in some region. It reminds be of big-O estimates of the number of primes less than a certain number n.
Tired of coming on here to be told for no reason I am yet again barred.
There had ought to be 3/2 as many primes in the Gaussians as there are in the integers, since every 4n+1 prime in the integers has two prime factors in the Gaussians, is what I meant by density.
I think one could define something like that density if one asked how many primes are less than the norm (or absolute value in integers). There might be more than 3/2 primes in the Gaussian integers less than a certain norm n since they are contained in a plane rather than just a line, but maybe not.
I just meant to give a rough idea that there are more, or at least seem to be.
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Well, it is very interesting that ideals somehow manage to recover unique factorization (though only in some limited cases, according to Wildberger, he called them quadratic instances, or something like that), so now the aim has to be to produce a specific example of how ideals managed to recover uinique factorization. After that, it is on to Carmichael numbers and the understanding of how they have anything to do with ideals. That is the plan anyway and as far as I can see right now. Something always gets in the way of direct progress.
I don't know how ideals do that either. That does seem like a major justification for considering them.
Ideals seem to eliminate the difference between 4n+1 primes and 4n+3 primes. That is, every ideal generated by a prime is a prime ideal regardless of its type to begin with. This is really cool, even if it only provides some unique factorization domains. The fact that any part of unique factorization has been recovered has got to be one of the greatest acheivements of mathematics.Quote:
Originally Posted by YesNo
The Gaussian integers are a unique factorization domain already. So, the 4n+1 or 4n+3 primes are not a problem. We just get different primes than we might have expected in the Gaussian integers.
But the integers with the sqrt(-5) are not a unique factorization domain. So here is where the ideals should help, but I don't see how at the moment.
In red, are you talking about something like 2+i and 2-i as primes?Quote:
Originally Posted by YesNo
There is a bit of mystery here. I am wondering why if the Gaussians are a UFD is there more than one way to factor numbers such as 6 or 5 within it? I can almost trust I am overlooking something.
Anyway, key things get remembered. Another one is
Quotinet rings divided through by a maximal ideal produce a field.
Since the field seems to consist of only 0 and 1, I can't see yet why that is so important, but it seems to be.
Here is another key to hold onto:
Ideals are to rings as normal subgroups are to groups.
That shouts: Go study groups, doesn't it?
I hope that is correct. There is a lot of talk of cosets, too, and I think that may be more group theory in spite of the name recalling set theory.
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My experience with being slow tells me a clear understanding of ideals is in front of my face, unrecognized, while I acclimate my brain to something new. All these little brealkthroughs will eventually amount to sort of an ephiphany.
The idea of principal ideals is pretty clear--they are just multiples. Like rings, ideals need 0 and 1 in the set--maybe some unit that stands in for 1, since the set of even numbers, for instance, cannot have a 1 in it.
Ideals are presented in additive terminology, though they have multiplicative properties.
I think this idea of "splitting," I keep running into refers to non-commutivity. I think this splitting is related to factorization problems. My present guess is that ideals manage to bypass this problem. I think this problem is related to the fact that 4n+3 primes are also primes in the Gaussians but 4n+1 primes are not, as they can be factored into smaller factors.
I hope I am not too amiss here. I am trying to put my collage together.
Up to units, there is only one way to factor 6 or 5 in the Gaussian integers. For example one can factor 6 = (3)(2) in Z. But one can also factor it as (-3)(-2). One could also reorder the factors as (2)(3). These are all different factorizations, but they are not what unique factorization tries to capture as an idea. The unique factors do not depend upon the order of the factors, nor do they depend on whether one can multiply the factors by 1 and get a different set of factors (associates). Again using 6 in Z, we can write 6 = (2)(3) = (1)(2)(3) = (-1)(-1)(2)(3) = (-2)(-3). In the Gaussian integers there are four units, not two as in Z: 1, -1, i, -1. Their norms are all 1. That challenges our normal intuition about what a unit should be (not just 1 or -1) and what an associate factor would be (not just multiplying the factor by -1).Quote:
Originally Posted by desiresjab
I am not clear about all of this either and I am finding it interesting to get a better understanding. In the Wikipedia article, https://en.wikipedia.org/wiki/Unique...ization_domain , there is a class chain:Quote:
Originally Posted by desiresjab
commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
There should be examples of non-unique factorization in structures to the left of "unique factorization domains" but none to the right. There should also be examples of a unique factorization domain that is not a principal ideal domain which is where the prime ideal questions we are discussing seem to be most important. But I am still unclear about how to formulate the questions.
Speaking of prime factorization in the Gaussians, you must be right. But in the Complex numbers is 2 even a prime? I can factor it as
(1+i)(1-i).
5 =(2+i)(2-i). Those are Gauusian integers, are they not (for a and b are both integers)? I believe those factors are not units, either. Does this make 5 not a prime in Gaussian integers. I believe no 4n+1 prime is a prime in the Gaussian integers, but I could be confusing Gaussians with the Complex numbers in general.
Now for 6 I really do not understand why 2x3 would be a prime factorization in the Gaussian integers, since I do not even believe 2 is a prime in that set. Isn't this the prime factorization?
(3)(1+i)(1-i)=6
What am I not getting about units for asking this question?
I have been looking around for an example of a non principal ideal. Of course it has been sitting in front of my nose.
All x+1 seem to form a non principal ideal. A non principal ideal is the kind I suspect Carmichael ideals to be.