The complex numbers are nice in two ways. They are an 'algebraically closed' field, meaning that every polynomial equation with complex coefficients has a complex solution. And they are 'Cauchy complete' metric space, meaning that every Cauchy sequence converges.
We can get the complex numbers in two ways. We can start with the rational numbers and take their Cauchy completion. This gives us the real numbers. But these are not algebraically closed. So we can take their algebraic closure. The result is the complex numbers, which is still Cauchy complete.
Or, we can start with the rational numbers and take their algebraic closure. This gives us the 'algebraic numbers'. There's a way to define a nice metric on these, but the resulting metric space is not Cauchy complete. To fix that, we can take its Cauchy completion. The result is the complex numbers, which is still algebraically closed.
In the first route I used the usual metric on rational numbers. But what if we use one of the p-adic metrics?
We can start with the rational numbers and take their Cauchy completion using the p-adic metric. This gives us the p-adic numbers. But these are not algebraically closed. So then we can take their algebraic closure. There's a nice metric on it, but it's 𝑛𝑜𝑡 still Cauchy complete.
So we can take the Cauchy completion 𝑎𝑔𝑎𝑖𝑛. You may feel sort of pessimistic right around now... but this time the resulting field 𝑖𝑠 algebraically closed, and of course Cauchy complete by definition. So yay, we're done! 🎉
The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬
@johncarlosbaez The amazing thing about complex numbers, at least for me, was how although a real function can be differentiable, it's derivative need not be (and all sorts of other patterns). But once a complex function is differentiable once it is infinitely differentiable. And has all sorts of other good things like Taylor series, and singularities that explain Taylor series radii, and more.
@dearlove - yes, this is the magic of the complex numbers. In real analysis, everything you want, you have to pay for. In complex analysis, a lot of wonderful stuff comes for free. Like: there's a differential equation that every differentiable function obeys! 😮
@johncarlosbaez Is there an accessible (I have a decent but dated first degree in maths, but that's mostly it) reference to that differential equation theorem?
@dearlove - if you have a differentiable function f on the complex numbers, its derivative in the y direction must be i times its derivative in the x direction, because the derivative is linear and the y direction is i times the x direction. So we get the Cauchy-Riemann equation:
[ \frac{\partial f}{\partial y} = i \frac{\partial f}{\partial x} ]
Most people take this and write 𝑓=𝑢+𝑖𝑣 and break it in two equations to make it harder to understand:
@johncarlosbaez Thanks. I hadn't thought of the CR equation (which, yes, I'm fairly sure most if not all presentations I've seen have broken in two) in that way. Been a long time since I looked at that.
@johncarlosbaez What if we first go from the rational numbers to the algebraic numbers? Is there a p-adic metric on the algebraic numbers? If we metrically complete this do we then have to algebraically complete a second tume?
@OscarCunningham - great question! I'm not an expert on this stuff, but there seem to be many different ways to embed the algebraic numbers (the algebraic closure of ℚ) into the algebraic closure of the p-adic numbers ℚₚ. Any one of these gives a "p-adic metric on the algebraic numbers".
An obvious next question is whether the algebraic numbers, enbedded in the algebraic closure of ℚₚ in one of these ways, is actually dense. If so, taking their metric completion gives the metric completion of ℚₚ, called ℂₚ, which is also algebraically closed.
@johncarlosbaez That's fascinating! Unfortunately I've only had limited exposure to the p-adic numbers, so I can't entirely appreciate this, but you've definitely piqued my interest.
"The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬"
Is this something someone has actually proven or is it just something that people assume?
Because when I took the seminar on this (circa 1982) , the word then was that we know very little about Ωₚ (the ultimate algebraically closed Cauchy-completion of 𝐐ₚ), that so far as we know, it has nothing to do with the complex numbers and Koblitz's book (what we were using) had nothing further to say,
and it seems to me there's rather a lot of work to do to axiom-of-choice a consistent metric out of this. But maybe somebody's done it in the last 30 years.
"Two algebraically closed fields are isomorphic if and only if they have the same characteristic and transcendence degree (see, for example Lang’s Algebra X §1). An algebraically closed field of transcendence degree λ has cardinality λ+ℵ₀ . If κ > ℵ₀ , an algebraically closed field of cardinality κ also has transcendence degree κ. Thus, any two algebraically closed fields of the same characteristic and same uncountable cardinality are isomorphic."
This does the job because the field I was talking about, like the complex numbers, is algebraically closed, with characteristic zero, and cardinality the continuum.
I guess the idea is that the theory of algebraically closed fields can't do anything to distinguish or relate algebraically independent transcendentals,
which means if we have two extensions of 𝐐 with their respective transcendence bases having the same cardinality, ANY bijection at all between those bases can be extended into a field isomorphism,
... and if it so happens there's no way to have such an isomorphism respect convergence of sequences (i.e., f(lim xₙ)=lim f(xₙ)) w.r.t. ANY of the metrics, we just don't care.
Seems like something they ought to have known about in 1982.
OTOH it's a completely useless isomorphism, so maybe it just wasn't worth mentioning.
@wrog - "if we have two extensions of 𝐐 with their respective transcendence bases having the same cardinality, ANY bijection at all between those bases can be extended into a field isomorphism".
Right. But using this to get an isomorphism between ℂ and ℂₚ requires "choosing" transcendence bases for each of these fields. And apparently this can only be done using the axiom of choice, not any actual construction.
and "choosing" every element of the (uncountably large) bijection
I knew that part. (ANY pulling of uncountably large rabbits from hats ⟹ AoC)
What I was missing was how to deal with the question of how you map some z that's the limit of a sequence in ℂ, to something appropriate in ℂₚ, where, even if the sequence were entirely rational numbers, it might not even converge in ℂₚ because the metric there is totally different and so you won't even have anything to choose from.
And the answer is if your z is algebraically independent of all of the z's you've considered thus far, then you just pick ANYTHING AT ALL that's algebraically independent of all of the previous f(z)'s and the metrics can go fuck themselves.
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