I don't understand this well enough at the moment. I will try to read the link again. I can see how Z[sqrt(-3)] can take advantage of ideals by calling a non-principal ideal an ideal number that doesn't exist in that set of integers and using that to get unique factorization, but that is as far as I see at the moment.
I am reporting from the site of another logjam. It surprised me to find out I did not know the proper procedure for generating the lattice on the Wolfram link:
http://mathworld.wolfram.com/Ideal.html
I can see too many ways to do it, I do not know which is the correct way. It seems to me from the theory I should be able to fill <2> and <1+i> in separately or as an active combination in <2, 1+i>. I can see how to get all the even numbers on the grid. I am not sure how to get the 1+i's beyond the inner group of 1+i, -1+i, etc. I may be more confused than I thought I was. I have cleared some path ahead but had to return to this.
Last edited by desiresjab; 02-13-2017 at 05:47 AM.
Since (1+i)*(1-i)=1+1=2, <1+i> should generate the same lattice as <2, 1+i>. To look at <2> separately, multiply it by all Gaussian integers in the visible part of the lattice by 2 and see where they lie. Certainly any even numbers on the real axis would be in the ideal as well as those on the complex axis. Some of the other points off the axes, but not all of them, should be present as well.
I am trying to see if I can treat <2, 1+i> like Cartesian coordinates.
Points on the a+bi axis (y-axis) are not clear to me. The red point in the upper right of the diagram looks like the Cartesian point (3, 3), but must have a different description in Gaussians. Since I do not even know what that point is I cannot figure out what to multiply by to get that point either.
Wait, I see my stupid mustake. The y-axis is not a+bi, it is the bi part alone of the expression. With that bit of foolishness out of the way, getting the points down the right way might be easier. Still don't know if I can generate them all the proper way.
Right. The x axis and the y axis would be the way one plots a real function, f(x) = y. The real axis and the complex axis would be how one plots a + bi. They have different names for these axes but both share in common the need to plot something in two dimensions.
There is a chapter in Birkhoff and MacLane on algebraic number fields that I hope will resolve some confusion I am having the Z[sqrt(-5)] and Z[sqrt(-3)] and how ideals help with unique factorization.
Okay, I have shown myself how to generate every point on the lattice through multiplication. These lattice diagrams may prove to be as germane to the study of ideals as Eisenstein's lattice diagram was for quadratic reciprocity. I want to make sure I get out of it everything there is to get, for I notice I can also take the additive approach and generate the same lattice, I believe. From any even number I can step 1+i or 1-i and get the remaining points, I mean. I still do not know if one way is the preferred way to see the lattice.
Last edited by desiresjab; 02-14-2017 at 07:04 AM.
I don't know if there is a preferred way to see this. Understanding something at all is all I aim for when looking at something I am unfamiliar with. However, finding different ways may lead to new results. That would imply a deeper understanding.
The lousy symbol processor on this site will not let me post again. I have a long post written. I do not feel like going through it to see what this system is objecting to, when I know it is the system again, not me.
This system uses brackets heavily for special functions, so whenever you use brackets in your post, especially in conjunction with a number, the system thinks you are trying to interfere with its propietary commands or something. Tired of it. These posts are hard to write, they include a lot of thinking.
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.
Last edited by desiresjab; 02-15-2017 at 12:05 AM.
Not sure what all this means. I feel certain key ideas have to be seen with perfect clarity, and this is one of them. Every point on this lattice has to be accounted for, using the generators given. There might be (almost certainly are) other generators that would fill in the exact same lattice, but I am only concerned with these two generators right now, what they do, and especially how they do it, whether in tandem or in isolation. See what I'm sane?
Unfortunately, what I had to leave out were the actual multiplications which showed how I arrived at each point. Frustratring.
(2-i)(1+i)=2+2i-i-i2=2+i+1=3+i
Last edited by desiresjab; 02-15-2017 at 12:06 AM.
Christ, to get 1+3i, multiply 1+i and 2+2i together. What is this garbage processor?
Last edited by desiresjab; 02-15-2017 at 12:30 AM.
Some legal multiplications produce results outside the lattice. Hmmm. Such as:
(1+i)(2+2i)=2+2i+2i+2i2=4i
Last edited by desiresjab; 02-15-2017 at 12:26 AM.
I think I know what you are trying to see. I do not know how to see it, either, but I believe I know what you are trying to see. The extension fields of √-5 and √-3. I cannot figure out what the devil they are talking about whent they speak of the primary difference between these two. It has to do with quotient fields and every integer of their qotient field already being there, at least for √-5, or some such thing.
Mathematicians have uglied up the quotient field thing real heavy with symbols. It may just mean this:
When you use an integer as your modulus over this complex field, your remainder often comes out with an imaginary piece. Well, technically it would always be there, but invisible when its constant was 0. A quotient field is somehow connected to this idea, I believe, but I do not know how and I certainly cannot prove or demonstrate it at this point.
The objects we are studying are becoming extremely abstract. It makes you appreciate the genius of the people who got there first with only their imaginations to guide them.
Last edited by desiresjab; 02-15-2017 at 01:00 AM.
Posting Permissions
Reply With Quote