Cosmology

[YesNo]

What seems queer is I do not believe I could generate points like (1+3i) and (3+i) with 2 by itself, but I can generate them all with (1+i), which is not surprising, but the fact that I can apparently generate all the evens with (1+i) as well, is, somewhat, at least. Of course, I still have to have 2 in there a bunch of times as a multiplier to accomplish this when using (1+i) as the generator.

Sorry I am having to post piecemeal what would have all been in one post.

You might try installing python through the anaconda distribution set. You can then create juypter notebooks and use mathjax which I think is close to LaTeX. We could share these notebooks.

The Gaussian integer 1+i should generate all the points on the lattice as you saw. Since 1+i divides 2, 2 should not generate all of the lattice points but only a subset of them. Although 4i is not on the portion of the lattice visible in the link, the lattice contains infinitely many points. It is just off the part that was shown.

[desiresjab]

Next, I sense we need to do some actual, ugly long-hand dividing in this territory to unlock some secrets and lighten some dark passage ways, but I do not even know how to start. We need the semblance of a problem to solve. Do you have a light?

[YesNo]

The way I would approach dividing is to construct the reciprocal and multiply. The reciprocal exists in the complex numbers but perhaps not in the Gaussian integers, so we can construct it.

For example suppose I wanted to divide 2 by 1+i. I would write that as 2/(1+i). But 1/(1+i) = 1(1-i)/(1+i)(1-i) = (1-i)/2. I multiplied the denominator (1+i) by its conjugate (1-i). That will give me an integer in the denominator. If I do that to the denominator, I have to do it to the numerator. That is why I multiplied 1/(1+i) by (1-i)/(1-i)=1. Now I can multiply 2 by (1-i)/2 and get 1-i.

That may not be what you are looking for. Being able to formulate a problem even if one cannot solve the problem is valuable work.

[desiresjab]

Thank you.

I think I am seeking the tie-in between addition and multiplication in ideals. I find something slightly peculiar there.

We know I can generate the Wolfram lattice with just 1+i. We know I cannot generate that lattice with merely the ideal of 2. But 2 should give me all the even points on the lattice, as I see it. From any even point I am able to reach the "odd," points simply by adding 1+i to this even value, in other words it seems just like combining the two ideals through addition, and it appears to work. I do not know if this method is valid. It seems like it would have to be. It seems like it represents that tie-in I am looking for between addition and multiplication in this relam.

Above all, ideals seem to be additive objects. I believe but cannot prove that I have now demonstrated this relationship. I have generated all the points on the lattice two different ways (separately through addition & multiplication), and I believe they are both valid, not just a coincidence. Perhaps I am wrong, but you see now what I am looking for, whereas you might not have before.

[desiresjab]

This link you once provided may have been excerpted from a class using a John Stillwell book on the history of mathematics, which I understand has a large section on abstract algebra.

http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

[desiresjab]

Though I am looking right at the examples in all these articles, I cannot yet see exactly how unique factorization has been recovered through ideals. I may be close, but not quite there yet. For me it may be a matter of tying together those definitions of ideals which depend on addition and those for muiltiplication.

When I get as close as I sense I am now, I begin to believe the job will be completed. I think we shall lay ideals bare soon.

[desiresjab]

I have to travel for a few days. Right when I would rather stay home and study I am forced to go on the road.

The difference between primes and irreducibles is still a problem to untangle. The article linked to last says the key idea in working with ideals is that the irreducibles and the primes do not match up in that realm. Now that is in the article almost word for word. As is the fact that the sum of two ideals is their GCD. Those are their words not mine. You know what I'm sane?

[YesNo]

Though I am looking right at the examples in all these articles, I cannot yet see exactly how unique factorization has been recovered through ideals. I may be close, but not quite there yet. For me it may be a matter of tying together those definitions of ideals which depend on addition and those for muiltiplication.

I am trying to make sense of that also, but I get distracted during the day.

I agree with what you said about adding terms from both <2> and <1+i> to get the ideal <2, 1+i>. In this case the ideal is principal and can be written as <1+i> in the Gaussian integers.

If one looks at Z[sqrt(-3)] there is an ideal that looks similar: <2, 1+sqrt(-3)>. Note that instead of i, we have sqrt(-3). This ideal is not principal because 2 does not divide into 1+sqrt(-3) like 2 was able to divide into 1+i in the Gaussian integers and get a Gaussian integer back. So what I understand we are to do is consider the ideal <2, 1+sqrt(-3)> as a new ideal number that we will add to Z[sqrt(-3)]. That is where the ideals help. That's how I understand it at the moment, but I might change my mind. This approach doesn't help with Z[sqrt(-5)] and that is where I am puzzled at the moment.

[desiresjab]

I am trying to make sense of that also, but I get distracted during the day.

I agree with what you said about adding terms from both <2> and <1+i> to get the ideal <2, 1+i>. In this case the ideal is principal and can be written as <1+i> in the Gaussian integers.

If one looks at Z[sqrt(-3)] there is an ideal that looks similar: <2, 1+sqrt(-3)>. Note that instead of i, we have sqrt(-3). This ideal is not principal because 2 does not divide into 1+sqrt(-3) like 2 was able to divide into 1+i in the Gaussian integers and get a Gaussian integer back. So what I understand we are to do is consider the ideal <2, 1+sqrt(-3)> as a new ideal number that we will add to Z[sqrt(-3)]. That is where the ideals help. That's how I understand it at the moment, but I might change my mind. This approach doesn't help with Z[sqrt(-5)] and that is where I am puzzled at the moment.

Of course the notion of 2 dividing into 1+i is odd anyway. It doesn't fit with our intuition. We could much easier appreciate how 1+i divides into 2. But with the peculiar definition of division in ideals we have 2 dividing 1+i.

I think better with my computer nearby. I will take a couple of days to try and bring everything together.

[YesNo]

I think I stated that wrong. 1+i divides 2 since (1+i)(1-i)=2. However, the ideal <2, 1+sqrt(-3)>in Z[sqrt(-3)] is not principal but it is prime or irreducible.

Using non-principal, but irreducible ideals is the way that ideals recovered unique factorization in those Dedekind domains that otherwise did not have unique factorization. There's something called the Fundamental Theorem of Ideal Theory that states that those Dedekind domains have unique factorization using ideals. I am trying to understand that proof. The Birkoff and MacLane book did not cover the proof and so I am reading Harry Pollard's "The Theory of Algebraic Numbers" to try to understand it better.

[desiresjab]

What I am finding is that whether a term is irreducible or prime seems to depend on the domain itself. For instance, in Z there does not seem to be any difference between irreducible and prime elements. One must always be aware if they are in an I.D, a P.I.D or a field to know these things. As I understand it, the definition of prime as we are familiar with it, is exactly the definition that suffices for irreducibles in ideal theory, but our familiar definition of primes is not sufficient for ideals.

There is also a theorem of Hilbert which converts any non-principal ideal into a principal one through multiplication of the ideal by a special Hilbert number. No details on this one yet.

Well, one thing we can say for sure is that a maximal ideal is always a prime ideal. Maximal ideals are fairly easy to get a handle on, thank Gog. I am unable to determine if maximal ideals include all of the prime ideals. I do not think so. So far, I believe there are other prime ideals which are not maximal but, of course, no maximal ideals which are not prime.

Sounds like you are really digging into the subject now. Some major insights must be on their way to you.

[YesNo]

Whatever insights I am getting have been discovered long ago. I am just sorting through the puzzle. I find this wikipedia page on Dedekind domains interesting at the moment: https://en.wikipedia.org/wiki/Dedekind_domain

Dedekind domains are integral domains in which there is unique factorization of ideals even though there may not be unique factorization of elements themselves. What that suggests to me is that there should be an example of a ring that is not a Dedekind domain, that is, where unique factorization of ideals does not work.

[YesNo]

Many insights are obvious after they are discovered. Then one wonders why one couldn't see them before. Here's one obvious insight that just recently became obvious to me.

Consider the greatest common divisor, g, of two integers, a and b. Given g, one can find two other integers x and y such that g = ax + by. Note the linear combination of a and b. If one considers all such combinations of a and b one has an ideal generated by a and b or <a, b>. That ideal is not principal, but because the integers are a principal ideal domain, one can find a single generator for <a, b> which would be g forming the principal ideal <g>.

I found on my bookshelves a translation of Dedekind's "Theory of Algebraic Integers" translated by John Stillwell. That insight about the gcd I mentioned above came from Stillwell's introduction, page 7. I forgot I even had that book and now for the first time, thanks to your discussion of these issues, desiresjab, I might actually finish reading it.

[desiresjab]

This is great. I urgently need all my concentration now to take steps in quicksand.

I believe Stillwell is the guy whose book (which you may be reading) was being used as the trext for the course taught here:

http://www2.math.ou.edu/~kmartin/nti/chap11.pdf

I am amazed by how many details I can pick up without the whole thing falling into place before my eyes. Sometimes even stubborn illusions are replaced by local understanding unable to force global epiphany.

Quite a few hurdles are terminological. When one finally receives the right information in the right form, the information usually sticks to the term like glue in the mind, but it often takes a long time for that event to happen--too long, for my tastes. I am dissatisfied with my mind's ability to gobble up these ideas like a young hen picking up corn. I would rather have a greater mind to work with than this old clunker.

* * * * *

There are three or four issues which if I could resolve, I would have a decent grip on ideals, unique factorization and field extensions, and precisely how they all fit together and what in concert they have acheived.

Another barricade is that much of ideal theory is expressed in group theoretical notation with groups leading the idea train. From the beginning ideals had deep connections with group theory and was developed with them in mind. I do not know that subject well enough to catch all the hints.

[YesNo]

The way I avoid quicksand is to sleep on it, stop reading, skim or switch to some other text when the one I'm reading becomes too difficult. Of course, that means I might never go back to the original text which is what happened with that Dedekind book long ago. I have only read Stillwell's introduction, but he has a very good style. The parts I understand are clear. The parts I don't understand it is probably my fault that I don't understand them.

Another insight that is now obvious to me is that when one extends the rationals Q with sqrt(-1) = i to get Q(i) one gets a field smaller than the complex numbers. It doesn't even contain all the real numbers. For example, it doesn't include any transcendental numbers such as pi or e because they aren't in Q and they can't be formed from a + bi where a and b are in Q. But when I see something written as a + bi I assume I am working in the complex numbers when actually I am working in a subfield of the complex numbers.