Cosmology

[desiresjab]

There seems to be some confusion even among the experts as to what is called what in the theory.

I think I am getting fairly close to a decent understanding. It really helps to work out these little inconsistencies and get everything in its proper place.

One thing that makes ideals so fun to study is that the word is not used elsewhere in mathematics with a gross of other meanings. A brand new word for a brand new idea. One could improve mathematics considerably by combing through an English dictionary and finding suitable terms to replace those that are overworked to the point of ambiguity.

[desiresjab]

Oooh! Oooh! I got something else straight that has been mystifying me, and which shows my guess was wrong about "splitting fields" being connected with 4n+1 and 4n+3 primes.

Splitting refers to factoring a polynomial all the way down to linear factors. Even if you factor it you have not split it unless all the factors are linear.

A Splitting Field involves something called Field Extensions, which are literally what they say. Mathematicians take a field like the rational numbers and "Adjoin," enough non-rational numbers to it to enable them to Split a particular ploynomial or class of polynomials into linear factors. The trick is to Adjoin just enough numbers to the original set to get the job done, instead of ending up with a much larger set which indeed gets the job of splitting done but also contains many superfluous elements. You want to adjoin the minimum number of elements to the set that enables splitting. This involves a lot of complex techniques but the basic idea is not that hard, though I imagine the idea of adjoinment in this fashion was quite revolutionary when it was first proposed. The seeds may have originated in group theory.

[YesNo]

I don't know much about splitting fields, but the idea of adding only what one needs to the base field in order to factor a polynomial over the rationals sounds interesting.

A pair of confusing terms for me is "prime" and "irreducible". I assume primes only exist in a unique factorization domain, otherwise what one gets are irreducibles which are as far as one can factor an object in the algebraic structure. At least there is a factorization even though it is not the only one.

[desiresjab]

I don't know much about splitting fields, but the idea of adding only what one needs to the base field in order to factor a polynomial over the rationals sounds interesting.

A pair of confusing terms for me is "prime" and "irreducible". I assume primes only exist in a unique factorization domain, otherwise what one gets are irreducibles which are as far as one can factor an object in the algebraic structure. At least there is a factorization even though it is not the only one.

Yes, one would think every irreducible would be a prime, but that maybe is not the case. I do not yet know enough to satisfy myself but I am running out of ideas of how to proceed. I have not seen a bit of use for primaries and semi-primes, but I suppose that lies ahead.

Once ideals are understood fully, that, I believe, is the major portion of higher arithmentic that occupied great minds in the 20th century and late 19th. Going beyond ideals may require a measure of inventiveness . A full understanding of Artin's work along with ideals would be much of what one needs. I am not shooting for Artin's work, though. He is too difficult. I would be nowhere near ready for Artin at this point.

[desiresjab]

Dr. Salomone talks about that in this video. This guy lectures at light speed, and I like that. Subjects other professors take 45 minutes to discuss he dispatches of in ten minutes. No wasted time.

According to him the only cases where irreducibility and primality do not coincide are cases that are special and "not nice." I notice such special cases are usually shoved back and relegated to a later timetable when one is supposedly more advanced. The class is here is Abstract Algebra II.

[YesNo]

I figure I won't understand anything fully, but some things will be understood enough. Eventually you should be able to read Artin's work. At the moment I don't think I would understand it either. But if we kept searching for clues, it should eventually make sense. I suspect it would take less time and effort to understand Artin than to understand Joyce's Finnegans Wake.

[desiresjab]

I figure I won't understand anything fully, but some things will be understood enough. Eventually you should be able to read Artin's work. At the moment I don't think I would understand it either. But if we kept searching for clues, it should eventually make sense. I suspect it would take less time and effort to understand Artin than to understand Joyce's Finnegans Wake.

And this is the road to Artin. It forks ahead, if anyone wants to go there. Never say never. The mathematician seeks isolated clues about the structure of the universe, the artist builds an artificial one out of the materials at hand. There is some connection. It is a long ways off for the mathematician.

The right junction could present itself. There may be a turnoff toward Brocard. For this reason squares and anything about them is good to pick up. There is so much juice of squares left in field extensions, I have to stay a while longer. I also need now to go back to that difficult paper you linked to a few weeks ago, and see if I can better understand the operation of multiplication of ideals presented there.

I am getting more secure with ideals, and the idea of a subgroup within a ring that is only reachable through the subgroup itself, which precisely corresponds to what an ideal is, I hope.

Rings are to ideals as normal subgroups are to groups. Does this mean the friendly rings referred to by Salomone are the normal subgroups of group theory, as opposed to some other kind of subgroup that is not normal, like a non-commutative one, perhaps? I am going to stop guessing, but it is an addictive habit, and seems to serve me even when I am wrong, by keeping me asking questions until I am sure about something. Right now I am not sure what I am supposed to try to be sure about next, which is less fun than knowing where you should look, which is less fun than suspecting where you should look. At the moment I neither know or suspect.

[desiresjab]

I hope my old pal Yes/No is all right. It is not like him to leave posts unanswered. I wanted to say that division of ideals is not so challenging as it seemed at first glance. They call the ideal that does the dividing larger (as in terms of a larger set), for theoretcially its elements have greater density along the number line than the ideal it is dividing, such as 2Z dividing 4Z. The first may have more "elements," per unit distrance, but we realize both sets are actually infinite so there is no real difference in their cardinality but only in their density of occurence along the number line. However, it is plain to see that 2Z has a smaller generator than 4Z of which 4Z is a multiple, so naturally 2Z can divide it evenly.

As for multiplication of ideals, I am still trying to locate a paper I recently glanced at which explains it.

If you multiply two elements from the ideal, the result ends up back in the ideal. The cool thing about ideals is that if you multiply one of its element times one of the elements in the ring that is not in the ideal, the result ends up back in the ideal, too. They call that "absorbing," multiplication. In the case of non-commutative ideals, multiplication will be absorbed from either the left or the right.

I now have to look at some details of multiplication, then actually multiply some ideals together to inspect the results, then I should be done with this part and ready to investigate the relationship of Carmichael numbers to ideals.

[desiresjab]

I myself must now be gone for a few days traveling. I will investigate ideals and multiplication before I go so I will have something to dwell on while I am away. May you all be here and in good health when I return.

[YesNo]

I think I missed your previous to last post. What you say makes sense to me. I don't know how ideals fit in with Carmichael numbers, but I heard they do in some way. I'll see if I can find out more about that.

Here's something on the multiplication of ideals that I thought was interesting:

This one shows that the product of two ideals is not their intersection, in particular, 2Z multiplied by 2Z is 4Z,but that is not the intersection of 2Z with itself: http://math.stackexchange.com/questi...e-intersection

That question also gives a definition of what multiplying two ideals means. Let I and J be ideals and ij the product of one element from I and one from J. Consider the set of all finite sums of these kinds of pairs. That would be the product of two ideas. It has to be finite since an infinite sum would likely take one out of the ring.

[desiresjab]

That is the problem. I do not see where 2Z or 4Z is finite. Element by element multiplication makes perfect sense for a finite number of elements. How do these two ideals (any two ideals) work out be finite? I do not see that.

[YesNo]

One restricts the sums to be over a finite number of products by definition. The definition makes it finite. So we have I = 2Z = {...-4,-2,0,2,4,...}. Suppose we want to multiply the ideal I by itself. We can construct a set that contains every product, 2m * 2n, where m and n are integers. This multiplies every element in I by another element in I. Now take any finite number of those products and add them together. This becomes an element in the ideal that is formed from the product of 2Z * 2Z. The smallest positive integer in that product of ideals would be 4 and that would be the generator of the ideal. This is what we would expect because we get 2Z * 2Z = 4Z = {...-8,-4,0,4,8,...}.

Regarding Carmichael numbers the Wikipedia article says that the idea of Carmichael numbers is extended to other algebraic structures through ideals. Ideals don't help us solve problems about Carmichael numbers in the regular integers. So, ideals help generalize the idea of Carmichael numbers to other algebraic structures: https://en.wikipedia.org/wiki/Carmichael_number

[desiresjab]

Thanks again. I have to revisit an article to see if I can establish for a fact that the sum of two ideals is simply their GCD. That is very curious. Well, let me see. How about a concrete example? If I took two numbers such as 19 and 12. When I add them, their sum of 31 would actually be 1, in that case. So, yes, that seems curious. It seems modular. I can see where cycling back to 1 occurs in a cyclic ring such as a modulus, but how does that apply here? Of course the GCD of 19 and 12 separately is 1, so in that regard it makes sense. The notion here seems "not usual," and I will have to think about it some more and read some more.

[YesNo]

I think one defines the sum of two ideals as what one would get if one added one element of an ideal to an element of another ideal. So if (19) is an ideal in Z and (12) is an ideal in Z, what would be in their sum would be elements like 19a + 12b for integers a and b.

Thinking about this as a greatest common divisor, since we know that the GCD of 19 and 12 is 1, we should be able to write an equation like this: 19x + 12y = 1 for some integers x and y.

But now think of 19x as some element in (19) and 12y as some element in (12). This matches up the result of the GCD operation and the sum of two ideals.

[desiresjab]

More is becoming clear but the muddled parts still drive me nuttier. There are quite a few items in the following article which confuse me. For instance, I do not "see," why or understand statelments like

Z[√-5] is already the full ring of integers of its quotient field Q(√-5). The examples they use are always √-3 and √-5, because larger ones quickly start to become unweildly. We would like to understand the difference in behavior in this arena of 4n+1 and 4n+3 primes, for apparently there is one, is about what I could get out of page four through page five of

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

Though I always pick up additional incidental details I do not fully understand or which, on the other hand, make perfect sense to me.