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On Computing Factors of Cyclotomic Polynomials
, 1993
"... For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ..."
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For odd squarefree n > 1 the cyclotomic polynomial n (x) satises the identity of Gauss 4 n (x) = A 2 n ( 1) (n 1)=2 nB 2 n : A similar identity of Aurifeuille, Le Lasseur and Lucas is n (( 1) (n 1)=2 x) = C 2 n nxD 2 n or, in the case that n is even and squarefree, n=2 ( x 2 ) = C 2 n nxD 2 n ; Here A n (x); : : : ; D n (x) are polynomials with integer coecients. We show how these coef cients can be computed by simple algorithms which require O(n 2 ) arithmetic operations and work over the integers. We also give explicit formulae and generating functions for A n (x); : : : ; D n (x), and illustrate the application to integer factorization with some numerical examples.
Computing Aurifeuillian factors
 In Computational Algebra and Number Theory, Mathematics and its Applications Vol. 325
, 1995
"... Abstract. For odd squarefree n> 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) 2 ± nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require ..."
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Abstract. For odd squarefree n> 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) 2 ± nxDn(x) 2 of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require O(n 2) arithmetic operations over the integers. Also, there are explicit formulas and generating functions for Cn(x) and Dn(x). This paper is a preliminary report which states the results for the case n = 1 mod 4, and gives some numerical examples. The proofs, generalisations to other squarefree n, and similar results for the identities of Gauss and Dirichlet, will appear elsewhere. 1.
Harald Bohr The Football
"... The air buzzed with anticipation as the football team crowded excitedly into the lecture hall. The country’s top halfback was about to defend his Ph.D. thesis in mathematics! It soon became apparent that the proceedings were a mere formality, as the candidate’s dissertation on summability methods fo ..."
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The air buzzed with anticipation as the football team crowded excitedly into the lecture hall. The country’s top halfback was about to defend his Ph.D. thesis in mathematics! It soon became apparent that the proceedings were a mere formality, as the candidate’s dissertation on summability methods for divergent Dirichlet series was a masterful piece of work. This scenario is no fantasy from a 1990s television sitcom; it is a true story. The place was Copenhagen, the year was 1910, and the sport was “football ” as the word is understood internationally (“soccer ” in American lingo). The star halfback played in the 1908 Olympics on Denmark’s silvermedal football team, a team that is still in the record books [21, p. 172] for the most goals scored in a single game. (Denmark defeated France by the lopsided score of 17 to 1.) The dissertation title was Contributions to the Theory of
Representations of Finite Groups: A Hundred Years, Part I
"... Mathematical ideas in any subject area are often discovered and developed over a period of time, so it is usually not possible to assign a specific date to a discovery. But in a few cases a discovery may ..."
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Mathematical ideas in any subject area are often discovered and developed over a period of time, so it is usually not possible to assign a specific date to a discovery. But in a few cases a discovery may
Introduction to model theory
"... This course is an introduction in two senses. First, it is for people who haven’t studied model theory before, though I trust most people in the class ..."
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This course is an introduction in two senses. First, it is for people who haven’t studied model theory before, though I trust most people in the class
ON FINITE ARITHMETIC GROUPS
"... Abstract. Let F be a finite extension of Q, Qp or a global field of positive characteristic, and let E/F be a Galois extension. We study the realization fields of finite subgroups G of GLn(E) stable under the natural operation of the Galois group of E/F. Though for sufficiently large n and a fixed a ..."
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Abstract. Let F be a finite extension of Q, Qp or a global field of positive characteristic, and let E/F be a Galois extension. We study the realization fields of finite subgroups G of GLn(E) stable under the natural operation of the Galois group of E/F. Though for sufficiently large n and a fixed algebraic number field F every its finite extension E is realizable via adjoining to F the entries of all matrices g ∈ G for some finite Galois stable subgroup G of GLn(C), there is only a finite number of possible realization field extensions of F if G ⊂ GLn(OE) over the ring OE of integers of E. After an exposition of earlier results we give their refinements for the realization fields E/F. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups. 1.