The Arithmetic of Algebraic Numbers: An Elementary Approach (original) (raw)
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Formalized Mathematics, 2016
Summary This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial ring of ℚ[x] turns to be a field.
Fields of Algebraic Numbers Computable in Polynomial Time. I
Algebra and Logic, 2020
It is proved that the field of complex algebraic numbers has an isomorphic presentation computable in polynomial time. A similar fact is proved for the ordered field of real algebraic numbers. The constructed polynomially computable presentations are based on a natural presentation of algebraic numbers by rational polynomials. Also new algorithms for computing values of polynomials on algebraic numbers and for solving equations in one variable with algebraic coefficients are presented. * Supported by RFBR (project No. 17-01-00247) and by the RF Ministry of Science and Higher Education (state assignment to Sobolev Institute of Mathematics, SB RAS, project No. 0314-2019-0002). * * The work was carried out at the expense of the subsidy allocated to Kazan (Volga Region) Federal University for the fulfillment of the state assignment in the sphere of scientific activity, project No. 1.13556.2019/13.1.
The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting the postsecondary success of students majoring in STEM fields. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. The present study investigated the connections participants formed between algebraic procedures and basic number properties in the context of rational expressions. An assessment, given to 107 undergraduate students in precalculus, contained three pairs of closely matched algebraic and numeric rational expressions with the operations of addition, subtraction, and division. The researcher quantitatively analyzed the distribution of scores in the numeric and algebraic context. Qualitative methods were used to analyze the strategies and errors that occurred in the participants‟ written work. Finally, task-based interviews were conducted with eight participants to reveal their mathematical thinking related to numeric and algebraic rational expressions. Statistical analysis using McNemar‟s test indicated that the undergraduate participants' abilities related to algebraic rational expressions and rational numbers were significantly different, although serious deficiencies were noted in both cases. A small intercorrelation was found in only one of the three pairs of problems, suggesting that the participants had not formed connections between algebraic procedures and basic number properties. The analysis of the participants' written work revealed that the percent of participants who consistently applied the same procedure in the numeric and algebraic items of Problem Sets A, B, and C were 56%, 47%, and 37%, respectively. Correct strategies led to fewer correct solutions in the algebraic context because of a diverse collection of errors. These errors exposed a lack of understanding for the distributive and multiplicative identity properties, as well as the mathematical ideas of equivalence and combining monomials. These fundamental mathematical ideas need to be better developed in primary and secondary education. At the post-secondary level, these ideas should serve as the foundation for interventions that are designed to support underprepared students. The results of the interviews were consistent with the quantitative analyses and the qualitative examination of the strategies used by the participants. The findings in all three areas of the study point to a disconnect between numeric and algebraic contexts in the participants‟ thinking.
Algebraic Rational Expressions in Mathematics
2014
Algebraic rational expressions are a necessary component of the mathematics course in primary education. Hence, the need for appropriate methodical elaboration that enables enhanced acquisition of this abstract matter, which is the basis for improved adoption of numerous content areas in secondary education. This paper attempts to provide methodical guidelines for adoption of algebraic rational expressions with special attention to the possible intra-disciplinary integration with theory of numbers, geometry and writing numbers in expanded form.
Application of Algebraic Number Theory to Rational Theory
International Journal of Advanced Research in Science, Communication and Technology, 2023
This paper contained some notations connected with algebraic number theory and indicates some of its applications in the Gaussian field namely K(i) = (−1)..
Arithmetic with real algebraic numbers is in NC
1990
Abstract We describe NC algorithms for doing exact arithmetic with real algebraic numbers in the sign-coded representation introduced by Coste and Roy [CoR 1988]. We present polynomial sized circuits of depth &Ogr;(log 3 N) for the monadic operations-&agr;, 1/&agr;, as well as &agr;+ r, &agr;· r, and sgn (&agr;-r), where r is rational and &agr; is real algebraic.
The attainment and retention of later algebra skills in high school has been identified as a factor significantly impacting students’ postsecondary success as STEM majors. Researchers maintain that learners develop meaning for algebraic procedures by forming connections to the basic number system properties. In the present study, the connections students form between algebraic procedures and basic number properties in the context of rational expressions was investigated. An assessment given to 107 undergraduate students in Precalculus that contained three pairs of closely matched algebraic and numeric rational expressions was analyzed. McNemar’s test indicated that the undergraduate students’ abilities related to algebraic rational expressions and rational numbers were significantly different, although serious deficiencies were noted in both cases. A weak intercorrelation was found in only one of the three pairs of problems, suggesting that the students have not formed connections b...
Real algebraic numbers and polynomial systems of small degree
Theoretical Computer Science, 2008
We present exact and complete algorithms based on precomputed Sturm-Habicht sequences, discriminants and invariants, that classify, isolate with rational points and compare the real roots of polynomials of degree up to 4. We have closed formulas for all isolating points. Moreover we combine these results with a simple version of rational univariate representation so as to isolate and compute the multiplicity of all common real roots of a bivariate system of integer polynomials of total degree ≤ 2. We present our implementation within synaps and we perform experimentation and comparison with all available software. Our package is 2-10 times faster, even when compared to inexact software or to sofware with intrinsic filtering.