Displaying similar documents to “Algorithms for function fields.”

A polynomial reduction algorithm

Henri Cohen, Francisco Diaz Y Diaz (1991)

Journal de théorie des nombres de Bordeaux

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The algorithm described in this paper is a practical approach to the problem of giving, for each number field K a polynomial, as canonical as possible, a root of which is a primitive element of the extension K / ℚ . Our algorithm uses the L L L algorithm to find a basis of minimal vectors for the lattice of ℝ n determined by the integers of K under the canonical map.

On computing subfields. A detailed description of the algorithm

Jürgen Klüners (1998)

Journal de théorie des nombres de Bordeaux

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Let ℚ ( α ) be an algebraic number field given by the minimal polynomial f of α . We want to determine all subfields ℚ ( β ) ⊂ ℚ ( α ) of given degree. It is convenient to describe each subfield by a pair ( g , h ) ∈ ℤ [ t ] × ℚ [ t ] such that g is the minimal polynomial of β = h ( α ) . There is a bijection between the block systems of the Galois group of f and the subfields of ℚ ( α ) . 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...

A fast algorithm for polynomial factorization over ℚ p

David Ford, Sebastian Pauli, Xavier-François Roblot (2002)

Journal de théorie des nombres de Bordeaux

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We present an algorithm that returns a proper factor of a polynomial Φ ( x ) over the p -adic integers ℤ p (if Φ ( x ) is reducible over ℚ p ) or returns a power basis of the ring of integers of ℚ p [ x ] / Φ ( x ) ℚ p [ x ] (if Φ ( x ) is irreducible over ℚ 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.

Topics in computational algebraic number theory

Karim Belabas (2004)

Journal de Théorie des Nombres de Bordeaux

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We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the system.

P-adic root isolation.

Thomas Sturm, Volker Weispfenning (2004)

RACSAM

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We present an implemented algorithmic method for counting and isolating all p-adic roots of univariate polynomials f over the rational numbers. The roots of f are uniquely described by p-adic isolating balls, that can be refined to any desired precision; their p-adic distances are also computed precisely. The method is polynomial space in all input data including the prime p. We also investigate the uniformity of the method with respect to the coefficients of f and the primes p. Our...