Recognizing 1-Euclidean Preferences: An Alternative Approach
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Line (geometry)
Real line
Euclidean algorithm
Euclidean domain
Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has trivial class group. This paper generalizes Harper's variation of Motzkin's lemma to Lenstra's concept of Euclidean ideals and then uses the large sieve to obtain growth results. It concludes that if a certain set of primes is large enough, then the ring of integers of a number field with cyclic class group has a Euclidean ideal.
Euclidean domain
Lemma (botany)
Euclidean algorithm
Ideal class group
Euclidean group
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Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has trivial class group. This paper generalizes Harper's variation of Motzkin's lemma to Lenstra's concept of Euclidean ideals and then uses the large sieve to obtain growth results. It concludes that if a certain set of primes is large enough, then the ring of integers of a number field with cyclic class group has a Euclidean ideal.
Euclidean domain
Lemma (botany)
Euclidean algorithm
Ideal class group
Euclidean group
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Euclidean domain
Euclidean algorithm
Principal ideal
Matrix (chemical analysis)
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Abstract In this paper we study number fields which are Euclidean with respect to functions that are different from the absolute value of the norm, namely weighted norms that depend on a real parameter c . We introduce the Euclidean minimum of weighted norms as the set of values of c for which the function is Euclidean, and we show that the Euclidean minimum may be irrational and not isolated. We also present computational results on Euclidean minima of cubic number fields, and present a list of norm-Euclidean complex cubic fields that we conjecture to be complete.
Euclidean domain
Euclidean shortest path
Maxima and minima
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Euclidean algorithm
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Students can use microcomputers to cut through algorithms and computations to gain mathematical insights. This approach is especially true for the Euclidean algorithm, so often used to find the greatest common divisor (GCD) of two positive integers. The Euclidean algorithm also yields continued fractions, at least far enough for students to find patterns and discover truths about numbers.
Euclidean algorithm
Greatest common divisor
Micro computer
Euclidean domain
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Euclidean domain
Euclidean algorithm
Greatest common divisor
SIGNAL (programming language)
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We give examples of Generalized Euclidean but not norm-Euclidean number fields of degree greater than 2. In the same way we give examples of 2-stage Euclidean but not norm-Euclidean number fields of degree greater than 2. In both cases, no such examples were known.
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The purpose of this paper is to study the concept of a greatest common divisor in such a way that it may be used as an enrichment topic at the senior high school level. In order to do this, the concept has been approached from both a theoretic and an applied point of view. In the theoretical areas, the algebraic structures involved are explored, along with the Euclidean algorithm, which provides a means for computing the greatest common divisor of two elements in a Euclidean ring and expressing it as a linear combination of these elements. In the more applied sections, such an algorithm is programmed for a computer. It is used in a BASIC program which will compute (a, b) in the ring of polynomial forms over a field and express it as a linear combination of a and b. Some other Euclidean rings, which the teacher might find instructive to explore with a class, are also discussed.
Greatest common divisor
Euclidean domain
Euclidean algorithm
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Line (geometry)
Real line
Euclidean algorithm
Euclidean domain
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A sufficient condition of Euclidean rings is given by polynomial optimization. Then, through computation, we give all norm-Euclidean square number fields, four examples of norm-Euclidean cubic number fields and two examples of norm-Euclidean cyclotomic fields, with the absolute of a norm less than 1 over the corresponding box, respectively.
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Square (algebra)
Euclidean shortest path
Euclidean algorithm
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