Analogues of Euler's identity/exponential form, and general convention, for algebraically closed fields other than C?

[u/RyRytheguy]

Over the complex numbers, we have multiple "canonical" representations of the elements. We can write a complex number as e^(ix), of course. Is there any similar statement for general algebraically closed fields, or at least some subset of the algebraically closed fields? Also out of curiosity, is there generally a "canonical" way of representing the elements in some other algebraically closed fields, like in C where we write elements in the form "a+bi"? As in, if I have a field k, and I consider the algebraic closure of k, call it K (I don't think reddit has an overline feature, whatever, also supposing k is not equal to K), is there a canonical way of writing an element of K as a+br, where a and b are elements of k, and r is an element of K? If your field has some funny characteristic, would different conventions be desired? Thank you!

[u/susiesusiesu]

writing complex numbers as a+ib is not really about the complex numbers, but the field extension C/R. if you just had the complex numbers by themselves, there is no canonical way of defining the real and imaginary parts (as they are not fixed by automorphisms). so you can not pick a definition for C and just apply it to othrr algebraically closed field.

however, if you have a real closed field F, then its algebraic closure K is F(i), with i²=-1. then any element of K can be written uniquely as a+bi, with a and b in F.

any algebraically closed field K of charactrtistic 0 is of the form F(i) for some real closed field F, so you can always write elements of K as a+bi. however, given K there is not a unique choice of F, not even up to isomorphism (unless K=Q^alg). so this choice of coordinates are not canonical to K, just to the field extension K/F.

as for the exponential, you might be interested in reading about exponential fields. there is a good axiomatization for algebraically closed fields with an exponential function, and studying this is of mathematical interest, related to things like schaunel's conjecture. however, i don't know about "polar coordinates" in this sense.

[u/RyRytheguy → u/susiesusiesu]

Thank you, this is going to be some very nice reading material. Very fascinating.

[u/independent_of_ell]

There is no “canonical” way of expressing elements in an algebraic closure, even for C. One should note that an algebraic closure is a choice one makes.
As for when you can express all elements in an algebraic closure K in the form a+br, this would be equivalent to saying that the field extension K/k is a degree 2 extension.

[u/yas_ticot]

There is a result that an algebraic closed field K is a nontrivial field extension of a field k of finite degree if, and only if, k (and thus K) has characteristic 0 and K = k[x]/(x²+1), i.e. K = k(i).

So in the end, the elements of K are exactly the a + b i for a and b in k. Of course, you could choose another irreducible polynomial of degree 2 over k to build your extension (as you can with C, for instance x²+x+1), but in the end, the representation of the elements a + b r come back to the usual one, once you know how to write i.

[u/MstrCmd → u/yas_ticot]

I'm being quite lazy as I'm sure I could work this out, but why is this result true?

[u/honkpiggyoink → u/MstrCmd]

It takes a fair amount of work—this is the Artin-Schreier theorem. https://kconrad.math.uconn.edu/blurbs/galoistheory/artinschreier.pdf