Minimality of topological matrix groups and Fermat primes

Our aim is to study topological minimality of some natural matrix groups. We show that the special upper triangular group ST⁺(n,F) is minimal for every local field F of characteristic ≠2. This result is new even for the field R of reals and it leads to some important consequences. We prove criteria for the minimality and total minimality of the special linear group SL(n,F), where F is a subfield of a local field. This extends some known results of Remus–Stoyanov (1991) and Bader–Gelander (2017). One of our main applications is a characterization of Fermat primes, which asserts that for an odd prime p the following conditions are equivalent: (1) p is a Fermat prime; (2) SL(p−1,Q) is minimal, where Q is the field of rationals equipped with the p-adic topology; (3) SL(p−1,Q(i)) is minimal, where Q(i)⊂C is the Gaussian rational field.