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Constructing class fields over local fields

Sebastian Pauli — 2006

Journal de Théorie des Nombres de Bordeaux

Let K be a 𝔭 -adic field. We give an explicit characterization of the abelian extensions of K of degree p by relating the coefficients of the generating polynomials of extensions L / K of degree p to the exponents of generators of the norm group N L / K ( L * ) . This is applied in an algorithm for the construction of class fields of degree p m , which yields an algorithm for the computation of class fields in general.

GiANT: Graphical Algebraic Number Theory

Aneesh KarveSebastian Pauli — 2006

Journal de Théorie des Nombres de Bordeaux

While most algebra is done by writing text and formulas, diagrams have always been used to present structural information clearly and concisely. Text shells are the interface for computational algebraic number theory, but they are incapable of presenting structural information graphically. We present GiANT, a newly developed graphical interface for working with number fields. GiANT offers interactive diagrams, drag-and-drop functionality, and typeset formulas.

A fast algorithm for polynomial factorization over p

David FordSebastian PauliXavier-François Roblot — 2002

Journal de théorie des nombres de Bordeaux

We present an algorithm that returns a proper factor of a polynomial Φ ( x ) over the p -adic integers p (if Φ ( x ) is reducible over p ) or returns a power basis of the ring of integers of p [ x ] / Φ ( x ) p [ x ] (if Φ ( x ) is irreducible over p ). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algorithm.

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