When I tossed this season’s A To Z open to requests I figured I’d get some surprising ones. So I did. This one’s particularly challenging. It comes from Gaurish Korpal, author of the Gaurish4Math blog.
A major field of mathematics is Algebra. By this mathematicians don’t mean algebra. They mean studying collections of things on which you can do stuff that looks like arithmetic. There’s good reasons why this field has that confusing name. Nobody knows what they are.
We’ve seen before the creation of things that look a bit like arithmetic. Rings are a collection of things for which we can do something that works like addition and something that works like multiplication. There are a lot of different kinds of rings. When a mathematics popularizer tries to talk about rings, she’ll talk a lot about the whole numbers. We can usually count on the audience to know what they are. If that won’t do for the particular topic, she’ll try the whole numbers modulo something. If she needs another example then she talks about the ways you can rotate or reflect a triangle, or square, or hexagon and get the original shape back. Maybe she calls on the sets of polynomials you can describe. Then she has to give up on words and make do with pictures of beautifully complicated things. And after that she has to give up because the structures get too abstract to describe without losing the audience.
Dedekind Domains are a kind of ring that meets a bunch of extra criteria. There’s no point my listing them all. It would take several hundred words and you would lose motivation to continue before I was done. If you need them anyway Eric W Weisstein’s MathWorld dictionary gives the exact criteria. It also has explanations for all the words in those criteria.
Dedekind Domains, also called Dedekind Rings, are aptly named for Richard Dedekind. He was a 19th century mathematician, the last doctoral student of Gauss, and one of the people who defined what we think of as algebra. He also gave us a rigorous foundation for what irrational numbers are.
Among the problems that fascinated Dedekind was Fermat’s Last Theorem. This can’t surprise you. Every person who would be a mathematician is fascinated by it. We take our innings fiddling with cases and ways to show an + bn can’t equal cn for interesting whole numbers a, b, c, and n. We usually go about this by saying, “Suppose we have the smallest a, b, and c for which this is true and for which n is bigger than 2”. Then we do a lot of scribbling that shows this implies something contradictory, like an even number equals an odd, or that there’s some set of smaller numbers making this true. This proves the original supposition was false. Mathematicians first learn that trick as a way to show the square root of two can’t be a rational number. We stick with it because it’s nice and familiar and looks relevant. Most of us get maybe as far as proving there aren’t any solutions for n = 3 or maybe n = 4 and go on to other work. Dedekind didn’t prove the theorem. But he did find new ways to look at numbers.
One problem with proving Fermat’s Last Theorem is that it’s all about integers. Integers are hard to prove things about. Real numbers are easier. Complex-valued numbers are easier still. This is weird but it’s so. So we have this promising approach: if we could prove something like Fermat’s Last Theorem for complex-valued numbers, we’d get it up for integers. Or at least we’d be a lot of the way there. The one flaw is that Fermat’s Last Theorem isn’t true for complex-valued numbers. It would be ridiculous if it were true.
But we can patch things up. We can construct something called Gaussian Integers. These are complex-valued numbers which we can match up to integers in a compelling way. We could use the tools that work on complex-valued numbers to squeeze out a result about integers.
You know that this didn’t work. If it had, we wouldn’t have had to wait for the 1990s for the proof of Fermat’s Last Theorem. And that proof would have anything to do with this stuff. It hasn’t. One of the problems keeping this kind of proof from working is factoring. Whole numbers are either prime numbers or the product of prime numbers. Or they’re 1, ruled out of the universe of prime numbers for reasons I get to after the next paragraph. Prime numbers are those like 2, 5, 13, 37 and many others. They haven’t got any factors besides themselves and 1. The other whole numbers are the products of prime numbers. 12 is equal to 2 times 2 times 3. 35 is equal to 5 times 7. 165 is equal to 3 times 5 times 11.
If we stick to whole numbers, then, these all have unique prime factorizations. 24 is equal to 2 times 2 times 2 times 3. And there are no other combinations of prime numbers that multiply together to give us 24. We could rearrange the numbers — 2 times 3 times 2 times 2 works. But it will always be a combination of three 2’s and a single 3 that we multiply together to get 24.
(This is a reason we don’t consider 1 a prime number. If we did consider a prime number, then “three 2’s and a single 3” would be a prime factorization of 24, but so would “three 2’s, a single 3, and two 1’s”. Also “three 2’s, a single 3, and fifteen 1’s”. Also “three 2’s, a single 3, and one 1”. We have a lot of theorems that depend on whole numbers having a unique prime factorization. We could add the phrase “except for the count of 1’s in the factorization” to every occurrence of the phrase “prime factorization”. Or we could say that 1 isn’t a prime number. It’s a lot less work to say 1 isn’t a prime number.)
The trouble is that if we work with Gaussian integers we don’t have that unique prime factorization anymore. There are still prime numbers. But it’s possible to get some numbers as a product of different sets of prime numbers. And this point breaks a lot of otherwise promising attempts to prove Fermat’s Last Theorem. And there’s no getting around that, not for Fermat’s Last Theorem.
Dedekind saw a good concept lurking under this, though. The concept is called an ideal. It’s a subset of a ring that itself satisfies the rules for being a ring. And if you take something from the original ring and multiply it by something in the ideal, you get something that’s still in the ideal. You might already have one in mind. Start with the ring of integers. The even numbers are an ideal of that. Add any two even numbers together and you get an even number. Multiply any two even numbers together and you get an even number. Take any integer, even or not, and multiply it by an even number. You get an even number.
(If you were wondering: I mean the ideal would be a “ring without identity”. It’s not required to have something that acts like 1 for the purpose of multiplication. If we insisted on looking at the even numbers and the number 1, then we couldn’t be sure that adding two things from the ideal would stay in the ideal. After all, 2 is in the ideal, and if 1 also is, then 2 + 1 is a peculiar thing to consider an even number.)
It’s not just even numbers that do this. The multiples of 3 make an ideal in the integers too. Add two multiples of 3 together and you get a multiple of 3. Multiply two multiples of 3 together and you get another multiple of 3. Multiply any integer by a multiple of 3 and you get a multiple of 3.
The multiples of 4 also make an ideal, as do the multiples of 5, or the multiples of 82, or of any whole number you like.
Odd numbers don’t make an ideal, though. Add two odd numbers together and you don’t get an odd number. Multiply an integer by an odd number and you might get an odd number, you might not.
And not every ring has an ideal lurking within it. For example, take the integers modulo 3. In this case there are only three numbers: 0, 1, and 2. 1 + 1 is 2, uncontroversially. But 1 + 2 is 0. 2 + 2 is 1. 2 times 1 is 2, but 2 times 2 is 1 again. This is self-consistent. But it hasn’t got an ideal within it. There isn’t a smaller set that has addition work.
The multiples of 4 make an interesting ideal in the integers. They’re not just an ideal of the integers. They’re also an ideal of the even numbers. Well, the even numbers make a ring. They couldn’t be an ideal of the integers if they couldn’t be a ring in their own right. And the multiples of 4 — well, multiply any even number by a multiple of 4. You get a multiple of 4 again. This keeps on going. The multiples of 8 are an ideal for the multiples of 4, the multiples of 2, and the integers. Multiples of 16 and 32 make for even deeper nestings of ideals.
The multiples of 6, now … that’s an ideal of the integers, for all the reasons the multiples of 2 and 3 and 4 were. But it’s also an ideal of the multiples of 2. And of the multiples of 3. We can see the collection of “things that are multiples of 6” as a product of “things that are multiples of 2” and “things that are multiples of 3”. Dedekind saw this before us.
You might want to pause a moment while considering the idea of multiplying whole sets of numbers together. It’s a heady concept. Trying to do proofs with the concept feels at first like being tasked with alphabetizing a cloud. But we’re not planning to prove anything so you can move on if you like with an unalphabetized cloud.
A Dedekind Domain is a ring that has ideals like this. And the ideals come in two categories. Some are “prime ideals”, which act like prime numbers do. The non-prime ideals are the products of prime ideals. And while we might not have unique prime factorizations of numbers, we do have unique prime factorizations of ideals. That is, if an ideal is a product of some set of prime ideals, then it can’t also be the product of some other set of prime ideals. We get back something like unique factors.
This may sound abstract. But you know a Dedekind Domain. The integers are one. That wasn’t a given. Yes, we start algebra by looking for things that work like regular arithmetic do. But that doesn’t promise that regular old numbers will still satisfy us. We can, for instance, study things where the order matters in multiplication. Then multiplying one thing by a second gives us a different answer to multiplying the second thing by the first. Still, regular old integers are Dedekind domains and it’s hard to think of being more familiar than that.
Another example is the set of polynomials. You might want to pause for a moment here. Mathematics majors need a pause to start thinking of polynomials as being something kind of like regular old numbers. But you can certainly add one polynomial to another, and you get a polynomial out of it. You can multiply one polynomial by another, and you get a polynomial out of that. Try it. After that the only surprise would be that there are prime polynomials. But if you try to think of two polynomials that multiply together to give you “x + 1” you realize there have to be.
Other examples start getting more exotic. They’re things like the Gaussian integers I mentioned before. Gaussian integers are themselves an example of a structure called algebraic integers. Algebraic integers are — well, think of all the polynomials you can out of integer coefficients, and with a leading coefficient of 1. So, polynomials that look like “x3 – 4 x2 + 15 x + 6” or the like. All of the roots of those, the values of x which make that expression equal to zero, are algebraic integers. Yes, almost none of them are integers. We know. But the algebraic integers are also a Dedekind Domain.
I’d like to describe some more Dedekind Domains. I am foiled. I can find some more, but explaining them outside the dialect of mathematics is hard. It would take me more words than I am confident readers will give me.
I hope you are satisfied to know a bit of what a Dedekind Domain is. It is a kind of thing which works much like integers do. But a Dedekind Domain can be just different enough that we can’t count on factoring working like we are used to. We don’t lose factoring altogether, though. We are able to keep an attenuated version. It does take quite a few words to explain exactly how to set this up, however.