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I wrote about generaliasations of real numbers and managed to follow one link to create characteristic.
What would it mean to “start the construction of the real numbers” with a different characteristic? Prime fields of positive characteristic have no ordering or metric, so I wouldn’t know how to define Dedekind cuts or Cauchy sequences. You did say “it makes more sense to get analogues of complex numbers” [emphasis added], by which I guess you are referring to algebraic closures? But I don’t see offhand how it makes any sense to get an analogue of the real numbers this way.
He may mean using one of the p-adic metrics and completing, topologically and algebraically…
But the p-adic numbers have characteristic zero.
I know. I can’t think of any other possible interpretation, though. (being rather flexible in interpreting things)
You’re right, Mike. By “more”, I meant “any at all, as far as I know, but without ruling out the possibility that somebody may think of something else”.
So yes, take an algebraic closure, then complete topologically (and then possibly take an algebraic closure again because the topological completion destroyed it?), to get a positive-characteristic analogue of the complex numbers.
Although I don’t know any way to decompose that final result into “real” and “imaginary” parts, I wouldn’t want to say that there is no way to do it. I am genuinely ignorant.
Incidentally, I had a nice phone conversation with a non-mathematical friend this evening about what I had written today. I said that it was a historical accident that we named the real numbers after one of their properties (that they are real, not imaginary) and we could equally well have named them after any of a number of other properties (which were the basis of my last edit): finite, located, standard, -adic, characteristic-, and commutative (I forgot infinite-precision). And then I had fun explaining what each of these meant to somebody who once took calculus but mostly only remembers high-school algebra.
I guess I was confused by something similar to Mike what referred to in #2: in the Idea section it says, “the completion of the set of rational numbers. Here we are using the usual ordering of rational numbers; other orderings will give the -adic numbers instead.” I’m not sure about those “other orderings” (ordered in what sense? and is the implication that a completion would lead to a complete ordered something else??).
Yeah, I think that’s not quite right either – the p-adic numbers aren’t ordered. R is both the Dedekind-completion of Q relative to the usual ordering, and the Cauchy-completion of Q relative to the usual metric. If you pick a different metric, you get the p-adic numbers, but I don’t think you can get them by picking a different ordering – I don’t even know whether there are any other orderings on Q.
There is only one ordered field structure on Q. It follows from the theorem that restricts all norms on Q to the standard norm and the p-adic norms. I don’t remember the name offhand.
other ordings will give the p-adic numbers? instead.
Typo in the first section of the article.
I do not understand the thing about the characteristic. Already rationals are characteristic zero, and, by definition, every ring containing the field of rationals, what agrees with the characteristic of the field in the case the ring is a field. In particular, p-adic numbers form also a field fo characteristic zero; the discussion above makes an impression that it is not so.
@ Todd
Yes, it should say “other topological structures”, rather than “other orderings”.
@ Zoran
There are two separate things that one might do: start with the rational numbers but use a topological structure of nonzero adicity, or start with a prime field of nonzero characteristic. (But in the latter case, I only know how to get an analogue of the complex numbers, rather than an analogue of the real numbers.)
I redid the intro to fix the mistakes. I decided that there was no real need to actually talk about adics there at all; they’re at the bottom.
@Toby: Todd means “ordered field structures”, which naturally induce a topology. The notion of an ordered field is significantly stronger than that of an ordered set.
Yes, that’s how I know that I must have been wrong to have written “other orderings”: there are no other orderings on that are compatible with the field structure. (Proof: by induction on the numerator and denominator, showing that every rational number which is positive in the usual ordering must be positive in any field-compatible ordering.)
Mike wrote:
What would it mean to “start the construction of the real numbers” with a different characteristic? […] You did say “it makes more sense to get analogues of complex numbers” [emphasis added],
And I replied:
You’re right, Mike. By “more”, I meant “any at all, as far as I know, but without ruling out the possibility that somebody may think of something else”. […]
Arguably, the only real characteristic- numbers are the rational ones, that is the elements of the finite prime field itself. After all, this has a natural topology; it’s just that this topology is discrete, so it’s already complete. But perhaps one should take the local field instead.
You can also have elements. Which construction can you do then ?
If you buy either of the ideas above, that the characteristic- real numbers are precisely the elements of or that they are all of the elements of , then either way any of the other elements of would be imaginary.
Thanks.
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