Let be an algebraic number field given by the minimal polynomial of . We want to determine all subfields of given degree. It is convenient to describe each subfield by a pair such that is the minimal polynomial of . There is a bijection between the block systems of the Galois group of and the subfields of . These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding...
We study the asymptotics conjecture of Malle for dihedral groups of order , where is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.
We prove that van Hoeij’s original algorithm to factor univariate polynomials over the rationals runs in polynomial time, as well as natural variants. In particular, our approach also yields polynomial time complexity results for bivariate polynomials over a finite field.
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