In this paper, we classify metacyclic p-groups of nilpotency class at least three up to isomorphism by giving a canonical presentation for each isomorphism class.
In this paper we classify all infinite metacyclic groups up to isomorphism and determine their non-abelian tensor squares. As an application we compute various other functors, among them are the exterior square, the symmetric product, and the second homology group for these groups. We show that an infinite non-abelian metacyclic group is capable if and only if it is isomorphic to the infinite dihedral group.
Let f (x) be an irreducible degree four polynomial defined over a field F and let K = F (α) where α is a root of f in some fixed algebraic closure F of F . Several methods appear in the literature for computing the Galois group G of f , most of which rely on forming and factoring resolvent polynomials; i.e., polynomials defining subfields of the splitting field of f . This paper surveys those methods that generalize to arbitrary base fields of characteristic 0. Further, we describe a non-resolvent method that determines if K has a quadratic subfield by counting the number of roots of f that are contained in K, and we also describe how to construct explicitly a polynomial defining a quadratic subfield. We end with a comparison of run times for the various algorithms in the case F is the rational numbers.
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