One’s complement. One of the advantages of two’s complement is that it lacks a negative zero, which is a complication you don’t need in integer arithmetic.
(It does have some utility in floating point, where it can represent an underflow and the sign thereof.)
And there are indeed an infinite number of roots of unity, all of which are just points on the unit circle in the complex plane.
0 is already a ring. 
~Max
Yeah, but what DPRK is saying is that {0} is also a ring.
Thanks! Although, back into my reduced posting hole I likely go, perhaps to pop up next again in several months, who knows? (Apparently, my last post before this thread was a one minor note in August 2018, and before that in March 2018)
And, yup, there’s an edit on that post, thereby restoring the Cosmic Indistinguishable Balance.
Let me get back to the zero ring {0}. It is a ring, actually called the terminal ring, the one that every ring has a unique homomorphism to. I recall one book, I have forgotten which, in which 1 != 0 is part of the definition. That’s reasonable for fields where you don’t expect a terminal object, but for rings is simply wrong and anathema to category theorists like me.
The simplest field has two elements 0 and 1 with 1 + 1 =0.
I suppose that it depends on why you’re talking about rings. If you’re viewing a ring as a context, and studying the relationships between different elements of a ring, then {0} is pretty useless, because there’s nothing interesting to say within that context. But if rings themselves, and the relationships between them, are what you’re studying, then the trivial ring {0} is probably just as important as the trivial number 1 or 0.
We were talking about definitions that let us make natural statements cleanly without corner cases before, and in that same vein, expanding on what Hari said, if one rules out the terminal ring as not really a ring, all kinds of statements suddenly acquire corner cases; e.g., “a ring modulo any set of elements is a ring… UNLESS 1 can be written as a sum of left-multiples of right-multiples of values in that set”.
There is the fairly natural notion stronger than a ring of an “integral domain”, though, which is a ring which can be embedded into a field. These are the same as commutative rings where 0 has the primality property that no product of nonzero values is zero (including the empty case that 1 isn’t 0).
People sometimes misdefine integral domains ignoring the empty case and calling the terminal one element ring an integral domain. This inclusion is just as much a wart as it’s exclusion from rings would be!
Regarding primes, if R is a ring (let’s assume commutative, and containing 1), and like number theorists we consider prime ideals rather than just prime numbers, then what is supposed to happen is that P is prime if and only if R modulo P is an integral domain. The example of the integers shows that (0) is a prime ideal in the ring of integers, for Z = Z/(0) is an integral domain. (Even though 0 is not normally counted as a prime number.) But, similar to the subject of this thread, (1) is not, and Z/(1) = the terminal one-element ring should not be an integral domain!
Relatedly, when people talk about “the field with one element”, the thing to keep in mind is that this is just a manner of speaking because there isn’t one. As reiterated above the smallest field has 2 elements, namely 0 and 1.
I don’t think so. Saying that -3 is a prime destroys the prime factorization theorem as well as -3 * -3 = 9. If you want to add negative integers, I’d say -1 is a prime and nothing else. -1 has factors of only itself and 1, and the unique prime factorization of -n is -1 * (the unique prime factorization of n)
I don’t see a problem with the textbook definition that the uniqueness of the factorization into primes is up to units, so when you match up the primes if q = up where u is a unit then everything is OK.
How else would you suggest to generalise it? Take, for example, Z[√2]. Let’s factor 7 = (3 + √2)(3 - √2). But also 7 = (2√2 + 1)(2√2 -1). The trouble is, we cannot really regard them as different factorizations, because if p = 3+√2 and q = 2√2-1, then p divides q, since q = (√2-1)p, but also q divides p, since p = (√2+1)q.
Make it even simpler. If -1 is a prime, then 6 = 23 = (-1)(-1)23 , and we’ve already lost uniqueness. Unless you say that (-1)*(-1) “doesn’t count”, and at that point, we might as well just say “up to units”.
All those statements of theorems that currently start “For any odd prime p” instead of “For primes p > 2” would have to be reworded to the latter (assuming that it fails for p=1, which would be a reasonable assumption if it fails for p=2 but is good for all other primes). I like the former wording because it sounds more clever, and it would be a shame to lose it.
Now you know why one is indeed the loneliest number.
I said earlier “the prime numbers without 1 all have much more in common with each other than they do with 1”, but let me make this more explicit, more formal, in such a way as illustrates why it is important to consider them as a class excluding 1.
Every prime number has exactly the same properties with respect to the structure of the positive integers, multiplication, and equality; for example, every prime p satisfies “For all a and b, if a * b is divisible by p, then either a is divisible by p or b is divisible by p”, no prime p satisfies “p * p = p”, every prime p satisfies “For all a, if p * p divides a * a, then p divides a”, no prime satisfies “There exists an x such that x * x = p”, etc. We can formalize this by noting that there are"automorphisms" of the positive integers preserving this structure by taking any prime number to any other prime number; that is, for any prime p and any prime q, there are back and forth functions f and g from positive integers to positive integers which undo each other [so f(g(n)) = n and g(f(n)) = n for all n] and which preserve multiplicative structure [so f(x * y) = f(x) * f(y) and similarly for g] such that f§ = q [and thus g(q) = p]. We can define these f and g straightforwardly, just by acting in the appropriately permuting way on numbers’ prime factorizations.
In this sense, all the prime numbers (excluding 1) are “isomorphic” to each other, with respect to the structure multiplication of positive integers. [Or just as well, we can do this extending to arbitrary integers].
However, 1 is not isomorphic to those prime numbers. It has different multiplicative properties. For example, 1 * 1 = 1, while p * p is never equal to p for any prime number.
Since all the values 2, 3, 5, 7, … have exactly the same multiplicative properties as each other, but 1 doesn’t have the same properties as them, it is very natural to restrict the definition of “prime number” to just 2, 3, 5, 7, … (which at any rate deserve some nice name in themselves), excluding 1.
Doesn’t that all just go back to unique prime factorization, though? All you’re really saying is why the unique prime factorization theorem is so important.
Sure. Yes, that’s what I’m doing. I’m saying unique prime factorization is important in a particular way which tell us some numbers are equivalent to other numbers in having exactly the same (multiplicative) properties, while some are not. And so these equivalence classes are natural to study, and they carve the primes out in one class and 1 in a different class.
I’m making formal the claim “The primes have much, much more in common with each other than they do with 1”. They in fact have EVERYTHING in common with each other, and not everything in common with 1, when “EVERYTHING” means properties of purely multiplicative structure.
Of course, it’s not just that there’s an isomorphism class for the primes. Every value falls in some isomorphism class. In multiplicative structure, 2 is isomorphic to 3 and 5, and there’s a separate isomorphism class containing 4 and 9 and 25, and there’s a separate isomorphism class containing 6 and 10 and 15, there’s a separate isomorphism class containing 12 and 18 and 20 and 45 and 50 and 75. And all of these are separate from an isomorphism class that contains 1 alone.
So all these classes might as well be considered natural concepts carving reality at the joints. And, yes, the one with the primes is particularly useful even beyond the rest. But all I really care to point out right now is that the one with the primes is separate from the one with 1. Every single multiplicative statement which is true about one prime is true about every other prime, but 1 falls outside of that.
Incidentally, a good exercise for anyone who hasn’t tried it before is to think about why factorizations into not-further-factorable values always exist uniquely. Most people seem to think this is much more automatic and obvious than it really is, and can’t give a good explanation of it.
For example, why couldn’t it be that there were distinct integers p, q, r, and s, all bigger than 1, all with no integer factors between 1 and themselves, satisfying the equation p * q = r * s?
One might say “Well, it IS true, and I think it’s obvious, so who are you tell me I’m wrong in finding it obvious?”. But I would say, if you cannot give a good explanation of why unique maximal factorization is true that pins down the relevant conditions that would let you also readily understand which other contexts outside of the integers this unique maximal factorization should and should not similarly generalize to, you do not really have a proper argument for why unique maximal factorization is true (since you aren’t able to give an explanation that highlights the appropriate causal factors, so to speak).
[*: By other contexts, I mean things like how we also have uniqueness of maximal factorizations in a suitable sense in the complex numbers with integer components, but not in, say, the natural numbers whose last digits are 1, nor in the complex numbers of the form a + b * sqrt(-5) for integer a and b]
Actually, about those complex numbers with integer components… It’s often pointed out that in that domain, 5 is not a prime, because 5 = (1+2i)(1-2i). But it’s also true that 5 = (2+i)(2-i). Do (1+2i) and (2+i) have some factor in common?