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Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 42 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
A Polynomial Time Nilpotence Test for Galois Groups and Related Results
"... We give a deterministic polynomialtime algorithm to check whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time is polynomial in size (f). Also, we generalize the LandauMiller solvability test to an algorithm that tests if Gal (f) is in Γd: this algori ..."
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We give a deterministic polynomialtime algorithm to check whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time is polynomial in size (f). Also, we generalize the LandauMiller solvability test to an algorithm that tests if Gal (f) is in Γd: this algorithm runs in time polynomial in size (f) and nd and, moreover, if Gal (f) ∈ Γd it computes all the prime factors of #Gal (f). 1
Upper Bounds on the Complexity of some Galois Theory Problems
, 2008
"... Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ Z[x] can be computed in P#P. Furthermore, the order can be approximated by a randomized polynomialtime algorithm with access to an NP oracle. For ..."
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Assuming the generalized Riemann hypothesis, we prove the following complexity bounds: The order of the Galois group of an arbitrary polynomial f(x) ∈ Z[x] can be computed in P#P. Furthermore, the order can be approximated by a randomized polynomialtime algorithm with access to an NP oracle. For polynomials f with solvable Galois group we show that the order can be computed exactly by a randomized polynomialtime algorithm with access to an NP oracle. For all polynomials f with abelian Galois group we show that a generator set for the Galois group can be computed in randomized polynomial time. 1