Cosmology

[desiresjab]

One thing we can be sure of: the concept of left and right ideals is applicable only for non commutitive rings. In a commutative ring such as a modulus ring, left and right ideals are the same, since x times r and r times x are not different. So any time they start tlking about left and right ideals or splitting, you know they are talking about some non commutitive object.

Everywhere I turn are statements I do not understand. All I can do is take them one at a time, putting them on the hold list until I can get to them. For instance, why and how there are exactly 21 different quadratic fields is still quite a mystery to me.

[desiresjab]

Since ideals at heart are instances of multiples, we always need to be able to see how any ideal is a multiple. Any time that becomes clear, we have understood the ideal. Even with Carmichael numbers the quest can be reduced to finding and understanding what this multiple is of. What is the generator and what does it leave in its tracks?

In the case of Complex numbers there have to be two generators, as I see it, one for the integral x-axis and one for the imaginary y-axis. Even if they do not show it in many diagrams, the y-axis is really the i-axis, 1i, 2i, 3i etc. An example would be the lattice diagram in the following link:

http://mathworld.wolfram.com/Ideal.html

[desiresjab]

In the case of Complex numbers there have to be two generators, as I see it, one for the integral x-axis and one for the imaginary y-axis...http://mathworld.wolfram.com/Ideal.html

I am not too sure about my statement here. I mean, I still think there are two generators (are they called units, or not?), but I am not sure they are assigned to each axis. (1+i) is not even on an axis.

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My oversight recently that not every Gaussian integer is a root of an equation, or something like that, was to due to a mix up with something I read. Even as I wrote it I knew it must be wrong, but I wrote it anyway because I thought I had read it on a long-time-without-sleep binge just prior to that. I must apologize for that embarrassment. I don't really know what I confused it with either.

* * * * *

On to the more interesting question of ways to factor 5 in the Complex numbers.

We already know √4+i)(√4-i)=4-√4 i+√4 i+1=5.

To write (2+i)(2-i)=4-2 i+2i i+1=5, is essentially a trivial change from √4 to 2, so is cheating and not a valid different factorization. But how about this?:

(1+2i)(1-2i) = 1+-2i+2i-4(i)² = 1+4 = 5

That is definitely a different factorization of 5. Are there others? Maybe. I have not validated your claim yet. All it takes is the above to show lack of unique factorization. I am simply curious if there are more, or infinitely many, as you said, I believe.

Hmmmm... I think it may be the case that these factorizations display a type of symmetry that is important later on, where exchanging a and b in the Gaussian integer does not change the result of an equation containing them. What we did in the factorizations above is exchange a and b.

But somehow I feel I can make it work for 7 as well, which is supposed to be a Gaussian prime. Let's take a look.

7 = √6+i)(√6-i)=6-√6 i+√6 i+1=7

Doesn't exchanging a and b have to work?

(1+√6i)(1-√6i)=1-√6i+√6i+6=7. Yes, it works.

Now I really am confused, I thought 7 was a Gaussian prime, but I have found two different ways of factoring it that seem distinct. The method should work on any Gaussian integer, in fact. These two factorizations do not seem trivially different. What is going on???????

[desiresjab]

Here is another factorization for 7.

(2+√3i)(2-√3i)=4+3=7.

Maybe those factors are not irreducible, so this factorization would not count. I do not know for sure. But there it is anyway, another factorization of 7. I could not find another one for 5.

[YesNo]

On to the more interesting question of ways to factor 5 in the Complex numbers.

We already know √4+i)(√4-i)=4-√4 i+√4 i+1=5.

There are infinitely many ways to factor 5 in the complex numbers. Let c be a complex number. Since the complex numbers are a field, 5/c = d is a complex number. Multiply both sides by c and get 5 = cd.

However, if one restricts attention to only the Gaussian integers https://en.wikipedia.org/wiki/Gaussian_integer, that is complex numbers like a + bi where a and b are in Z, then 5 should have a unique factorization into irreducibles (primes).

To write (2+i)(2-i)=4-2 i+2i i+1=5, is essentially a trivial change from √4 to 2, so is cheating and not a valid different factorization. But how about this?:

(1+2i)(1-2i) = 1+-2i+2i-4(i)² = 1+4 = 5

That is definitely a different factorization of 5. Are there others? Maybe. I have not validated your claim yet. All it takes is the above to show lack of unique factorization. I am simply curious if there are more, or infinitely many, as you said, I believe.

However, you seem to have two different factorizations above. I will have to check this further. Also, I am not sure why the Gaussian integers are a principal ideal domain which would make it be a unique factorization domain. So I will look up some proof for that as well.

Hmmmm... I think it may be the case that these factorizations display a type of symmetry that is important later on, where exchanging a and b in the Gaussian integer does not change the result of an equation containing them. What we did in the factorizations above is exchange a and b.

But somehow I feel I can make it work for 7 as well, which is supposed to be a Gaussian prime. Let's take a look.

7 = √6+i)(√6-i)=6-√6 i+√6 i+1=7

Doesn't exchanging a and b have to work?

(1+√6i)(1-√6i)=1-√6i+√6i+6=7. Yes, it works.

Now I really am confused, I thought 7 was a Gaussian prime, but I have found two different ways of factoring it that seem distinct. The method should work on any Gaussian integer, in fact. These two factorizations do not seem trivially different. What is going on???????

In the case of 7, note that √6 is not an integer, that is, an element of Z. Therefore √6+i and √6-i are not Gaussian integers, but complex numbers. In the field of complex numbers 7 is a unit and everything divides it, but not in the ring of Gaussian integers.

[desiresjab]

There are infinitely many ways to factor 5 in the complex numbers. Let c be a complex number. Since the complex numbers are a field, 5/c = d is a complex number. Multiply both sides by c and get 5 = cd.

However, if one restricts attention to only the Gaussian integers https://en.wikipedia.org/wiki/Gaussian_integer, that is complex numbers like a + bi where a and b are in Z, then 5 should have a unique factorization into irreducibles (primes).



However, you seem to have two different factorizations above. I will have to check this further. Also, I am not sure why the Gaussian integers are a principal ideal domain which would make it be a unique factorization domain. So I will look up some proof for that as well.



In the case of 7, note that √6 is not an integer, that is, an element of Z. Therefore √6+i and √6-i are not Gaussian integers, but complex numbers. In the field of complex numbers 7 is a unit and everything divides it, but not in the ring of Gaussian integers.

Oh, yes, that is correct, they are complex numbers, not Gaussian integers.

[desiresjab]

On your next to last comment in your last post, I comment: All principal ideals are bsed on the multiple concept, if I have my reading straight this time. I have a sneaking suspicion that there are ideals based on other properties than simply "being a multiple of." I have a hunch Carmichael numbers might be non-principal ideals. But I seem to be about 50-50 on the hunches these days.

[YesNo]

An ideal is also an additive subgroup. From that perspective, if there is more than one generator then one has to also consider not only the multiples of each of the generators, but also sums of anything generated from those two or more generators. One could generate the even numbers in the integers Z by using the principle ideal generated by 2 or the non-principal ideal generated by (2,4). However, those ideals are the same.

Your earlier observation about factoring 5 in two different ways still has me puzzled.

Edit: I think this resolves my earlier puzzlement:

The two factorizations of 5 provided earlier are the same up to units in the Gaussian integers. So unique factorization, up to units, still holds.

To see the significance of this, look at 10 in the integers. This factors as 10 = 2*5 but also as 10 = (-2)*(-5). Those are two different factorizations but not up to units since if I multiply (-2)*(-5) by 1 = (-1)*(-1) then (-2)*(-5)=2*5.

In the case of Gaussian integers there are four units: 1, -1, i, -i. If I multiply 1+2i by 1 = (-i)(i), I get (-i)(i)(1+2i) = i(-i+2) = i(2-i). If I multiply 1-2i by 1 = (-i)(i), I get (-i)(i)(1-2i) = -i(i+2) = -1(2+i). So with 1 = (i)(-i), I get 5 = (2+i)(2-i) = 1(2+i)(2-i) = (i)(-i)(2+i)(2-i) = i(1-2i)(2-i) = i(2-i)(1-2i) = (1+2i)(1-2i).

So the two factorizations are the same up to units.

[desiresjab]

All right, then.

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(√5+i)(√5-i)=5-√5 i+√5 i+1=6

(√1+i)(√1-i)(√2+i)(√2-i)=6

We know the second factorization is not fully commutable.

* * * * *

These factorizations are in complex numbers. We can be in complex numbers, it is a number system. Are we ever in Gaussian integers in that sense? I just see them as a complex number with the form xⁿ-1, not as a number system. Are we ever really in Gaussian integers, other than to examine one or a few? They are just a subset of the complex numbers, are they not, perhaps part of some quadratic field?

[YesNo]

If we restrict ourselves to using only elements from the integers then we are in the integers and can see what algebraic structure exists there (commutative ring). The same thing goes for the Gaussian integers. Just use only Gaussian integers. In the integers or the Gaussian integers one can talk about unique factorization and primes.

Now if we expand the set to include all the multiplicative inverses, we move from integers to rationals (or reals) or from Gaussian integers to complex numbers, then one loses the idea of prime and unique factorization, but one gets all those multiplicative inverses and the algebraic structure is different (field).

Alternatively one can think of elements in the set of Gaussian integers as not being complex numbers. This would justify having a different name for them even though they look the same and have many other properties of addition and multiplication in common.

[desiresjab]

Aren't Gaussian integers only a tiny slice (an infinite tiny slice) of the complex field? Only a few numbers are of the the form xⁿ-1. Or is that xⁿ+1? Christ!!!

Before, I was kind of thinking of Gaussian integers as just another name for Complex integers. I can see that is dead wrong. There are relatively so few Gaussian integers (in the sense of density) that I don't see what good they are. Anything recovered there would be for only a few numbers. In fact, the Gaussian integers seem really, really, really restricted.

I guess I will have to get used to tooling around in these various subsets and nail down their identities better. It turns out I do not yet know exactly where I am.

[desiresjab]

Wait! Wait! Wait! Wait! Wait! Where do I get my stupid notions from sometimes? I think from laziness, which looks at things too casually and without enough effort.

A Gaussian integer is simply a Complex number the coefficients of whose real and imaginary parts are both rational, in fact both integers. They are not that sparse, then. They will fall right where integers fall on the Cartesian plane. And didn't Gauss himself say that a complex number whose both parts are rational is an integer? He manages to get much out of nothing. These are merely the complex numbers that are just the regular integers, aren't they, the ones able to shed the extra clothing of a complex number and look like a normal integer--well, behave like one, too?

I am seeing it much better now, actually. I seem to need rescuing from misconstruction more often these days as the iron gets heavier.

[YesNo]

Yes, the Gaussian integers are those complex numbers where both the real and imaginary parts are integers, such as, 3 + 5i or 250 - 26345i.

I am studying Game Theory because my daughter is taking a class in it and she discusses it. Also, I've picked up a couple of books from the library on fractals. I want to see to what extent a market chart can be represented as a self-affine fractal and where that breaks down. The fractal should be random but the market chart is not according to socionomics although fractals can be used to model them.

[desiresjab]

Yes, the Gaussian integers are those complex numbers where both the real and imaginary parts are integers, such as, 3 + 5i or 250 - 26345i.

I am studying Game Theory because my daughter is taking a class in it and she discusses it. Also, I've picked up a couple of books from the library on fractals. I want to see to what extent a market chart can be represented as a self-affine fractal and where that breaks down. The fractal should be random but the market chart is not according to socionomics although fractals can be used to model them.

I love how complex functions act, I just do not like calculating them very much. It really is a job for computers, being more messy on every level. The Euclidian algorithm is more messy, too, and correspondingly easier to make a mistake in. I am not tempted to do many of these long-winded (ugly ) calculations. They are, in fact, almost precisely what I love to stay away from in math when I can and still understand. However, it is desirable to be able to distinguish Gaussian primes quickly from other numbers that might resemble one but are not. It could easily become necessary to get down in the mud of Gaussian integers and do the arithmetic in some context or other.

* * * * *

I have read a little on both Game Theory and Fractals. Can't help you, really. I do know stock prices were there right from the beginnings of the theory, however, as they were involved in some observations of a few pioneers, Lorentz among them, I believe. Now I remember, he happened to see a chart of historical cotton prices and noticed how similar it was to long term weather patterns.

[YesNo]

I think the Gaussian primes would be either primes from the integers with a remainder of 3 mod 4 or the factors in Gaussian integers of primes from the integers with a remainder of 1 mod 4. However, I don't know if that is true or not. That is, I can't think of how I would try to prove that at the moment, in particular prove that those two characteristics give all the Gaussian primes.