A massive family of complex mathematical symmetries called Lie groups can now be mapped directly onto the set of all rational numbers.
The Inverse Galois Problem has challenged mathematicians for nearly two centuries. Most experts believed these specific groups, known as PSL(n, q) and PSU(n, q), were too complex to be expressed using the simple rational numbers used in daily life. This work provides the first explicit recipe for creating an infinite series of these groups. Mapping these symmetries reveals deep connections between the geometry of continuous shapes and the rigid logic of arithmetic. This discovery brings the field closer to a universal dictionary that translates every possible symmetry into a solvable equation.
Simple Lie Groups of type An as Galois groups over Q
arXiv · 2604.27402
In this paper, we utilize our previous results on mod p monodromy of cyclic coverings of the projective line to realize a large series of groups of the form PSL(n, q) and PSU(n, q) as Galois groups over Q. We achieve for the first time a fully explicit infinite series of such groups where simultaneously the field can have arbitrarily large degree over the prime field and the group does not coincide with PGL(n, q) or PGU(n, q), respectively.