We know embarrassingly little about the symmetries of the roots of all polynomials with rational coefficients, or if you prefer, the absolute Galois group
In the title picture the roots of polynomials of degree
John Baez has an excellent post on the beauty of roots, including a picture of all polynomials of degree
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
The ‘missing’ symmetries, that is the Galois group
[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.
They discovered a concrete compact connected Hausdorff space
Let’s look at a finite field
An etale map
.
If we replace
Sadly, there’s a catch. Even though we have a concrete description of
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
As a side remark, some of these people have just published a book with the EMS Publishing House: Absolute arithmetic and

The basic idea is that the collection of all prime numbers,
Initially, one hoped that
But, over the last decade it became clear that
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