The quotation from Quanta was incomplete, and the Quanta article doesn't in fact claim that "totally real" means what "algebraic" actually means. Here's what the article actually says (added emphasis mine):
> A number is totally real if it satisfies a polynomial equation with integer coefficients that only has real roots.
No it's narrower than algebraic. If I have this right, the number is totally real if it's a root of a polynomial with integer coefficients, all of whose roots are real. So for example, the (real) cube root of 2 is not totally real, because x^3-2 has one real root and two complex roots. The idea of totally real is that not only is the root real, but all of its images under the polynomial's Galois group are also real.
Added: in fact the Quanta article gets this right. It says "A number is totally real if it satisfies a polynomial equation with integer coefficients that only has real roots." Also, Jordana Cepelowicz is a knowledgeable math writer and I believe she has a PhD in math.
It does seem like a bit of not totally accurate shorthand: totally real isn't a property of a number, but rather of the extension field where it lives. So sqrt(2) is a root of x^2-2 whose roots are +/- sqrt(2), and the splitting field of x^2-2 is Q(sqrt(2)) which is totally real. But sqrt(2) is also a root of x^4-4 whose roots are +/- sqrt(2) and +/- i sqrt(2), so the splitting field doesn't live inside the reals. The wiki article explains it better. I was unfamiliar with this idea before.
That's true but I think it's implicit here we're working with the algebraic extension generated by the number itself (which always contains the Galois conjugates).
I'm not sure if I'm using the jargon properly, but if r is the real cube root of 2, then I thought the extension generated by r is Q(r), which contains only reals. So (x^3 - 2) factors into (x-r) and the irreducible quadratic (x^2 + r*x + r^2). You have to do some extra thing to reach the splitting field, amirite?
> As always, Quanta Magazine reporting is just garbage. The name for such numbers is ‘algebraic’.
Any popularization is inevitably going to run into some inaccuracies—or else it's just re-publishing the technical papers—but the opinions of most mathematicians I know are that, far from being garbage, Quanta's reporting is distinctly better than most.
