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Topology and the symmetries of roots

We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group Gal(Q/Q).

In the title picture the roots of polynomials of degree 4 with small coefficients are plotted and coloured by degree: blue=4, cyan=3, red=2, green=1. Sums and products of roots are again roots and by a symmetry we mean a map on all roots, sending sums to sums and products to products and leaving all the green dots (the rational numbers) fixed.

John Baez has an excellent post on the beauty of roots, including a picture of all polynomials of degree 5 with integer coefficients between 4 and 4 and, this time, colour-coded by: grey=2, cyan=3, red=4 and black=5.

beauty of roots

In both pictures there’s a hint of the unit circle, black in the title picture and spanning the ‘white gaps’ in the picture above.

If we’d only consider the sub-picture of all (sums and products of) roots including the rational numbers on the horizontal axis and the roots of unity on the unit circle we’d get the cyclotomic field Qcyc=Q(μ). Here we know all symmetries: they are generated by taking powers of the roots of unity. That is, we know all about the Galois group Gal(Qcyc/Q).

The ‘missing’ symmetries, that is the Galois group Gal(Q/Qcyc) remained a deep mystery, until last week…

[section_title text=”The oracle speaks”]

On september 15th, Robert Kucharczyk and Peter Scholze (aka the “oracle of arithmetic” according to Quanta-magazine) arXived their paper Topological realisations of absolute Galois groups.

Peter Scholze

They discovered a concrete compact connected Hausdorff space Mcyc such that Galois extensions of Qcyc correspond to connected etale covers of Mcyc.

Let’s look at a finite field Fp. Here, Galois extensions of Fp (and there is just one such extension of degree n, upto isomorphism) correspond to connected etale covers of the circle S1.

An etale map XS1 is such that every circle point has exactly n pre-images. Here again, up to homeomorphism, there is a unique such n-fold cover of S1 (the picture on the left gives the cover for n=2).

.

If we replace Fp by the cyclotomic field Qcyc then the compact space Mcyc replaces the circle S1. So, if we take a splitting polynomial of degree n with coefficients in Qcyc, then there is a corresponding etale n-fold cover XMcyc such that for a specific point p in Mcyc its pre-images correspond to the roots of the polynomial. Nice!

Sadly, there’s a catch. Even though we have a concrete description of Mcyc it turns out to be a horrible infinite dimensional space, it is connected but not path-connected, and so on.

Even Peter Scholze says it’s unclear whether new results can be proved from this result (see around 39.15 in his Next Generation Outreach Lecture).

Btw. if your German is ok, this talk is a rather good introduction to classical Galois theory and etale fundamental groups, including the primes=knots analogy.



[section_title text=”the imaginary field with one element”]

Of course there’s no mention of it in the Kucharczyk-Scholze paper, but this result is excellent news for those trying to develop a geometry over the imaginary field with one element F1 and hope to apply this theory to problems in number theory.

As a side remark, some of these people have just published a book with the EMS Publishing House: Absolute arithmetic and F1-geometry



The basic idea is that the collection of all prime numbers, Spec(Z) is far too large an object to be a terminal object (as it is in schemes). One should therefore extend the setting of schemes to so called F1-schemes, in which Spec(Z) is some higher dimensional object.

Initially, one hoped that Spec(Z)/F1 might look like a curve, so that one could try to mimick Weil’s proof of the Riemann hypothesis for curves to prove the genuine Riemann hypothesis.

But, over the last decade it became clear that Spec(Z)/F1 looks like an infinite dimensional space, a bit like the space Mcyc above.

I’ll spare this to a couple of follow-up posts, but for now I’ll leave you with the punchline:

The compact connected Hausdorff space Mcyc of Kucharczyk and Scholze is nothing but the space of complex points of Spec(Qcyc)/F1!

Published in absolute geometry number theory