It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I. S. Luthar and I. B. S. Passi which sometimes permits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove wit...

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if p > (d2 − 3d+ 4)2, then there is no complete mapping polynomial f in Fp[x] of degree d ≥ 2. For arbitrary finite fields Fq, a similar non-existence result is obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f .

We assume the generalized Riemann hypothesis and prove an asymptotic formula for the number of primes for which Fp can be generated by r given multiplicatively independent numbers. In the case when the r given numbers are primes, we express the density as an Euler product and apply this to a conjecture of Brown–Zassenhaus (J. Number Theory 3 (1971), 306–309). Finally, in some examples, we compa...

We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients absolute groups. Assuming well-known conjectural description these groups, we show that has PBW property. Hence in particular it is a Koszul algebra. This answers positively conjecture by Positselski this case. also provide an analogous unconditional result about Pythagorean fields. Moreover, establish some res...