Toward the Unification of Physics and Number Theory
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Algebraic Numbers As Product of Powers of Transcendental Numbers
S S symmetry Article Algebraic Numbers as Product of Powers of Transcendental Numbers Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic; [email protected]; Tel.: +42-049-333-2801 Received: 17 June 2019; Accepted: 4 July 2019; Published: 8 July 2019 Abstract: The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C[x], and the problem of finding zeros in Q[x] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P(T)Q(T). In this paper, we generalize this result by showing the existence of algebraic Q (T) Qn(T) numbers which can be written in the form P1(T) 1 ··· Pn(T) for some transcendental number T, where P1, ... , Pn, Q1, ... , Qn are prescribed, non-constant polynomials in Q[x] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of zw when z and w are transcendental. Keywords: Baker’s theorem; Gel’fond–Schneider theorem; algebraic number; transcendental number 1. Introduction The name “transcendental”, which comes from the Latin word “transcendˇere”, was first used for a mathematical concept by Leibniz in 1682. Transcendental numbers in the modern sense were defined by Leonhard Euler (see [1]). A complex number a is called algebraic if it is a zero of some non-zero polynomial P 2 Q[x]. Otherwise, a is transcendental. Algebraic numbers form a field, which is denoted by Q. The transcendence of e was proved by Charles Hermite [2] in 1872, and two years later Ferdinand von Lindeman [3] extended the method of Hermite‘s proof to derive that p is also transcendental. -
Integers, Rational Numbers, and Algebraic Numbers
LECTURE 9 Integers, Rational Numbers, and Algebraic Numbers In the set N of natural numbers only the operations of addition and multiplication can be defined. For allowing the operations of subtraction and division quickly take us out of the set N; 2 ∈ N and3 ∈ N but2 − 3=−1 ∈/ N 1 1 ∈ N and2 ∈ N but1 ÷ 2= = N 2 The set Z of integers is formed by expanding N to obtain a set that is closed under subtraction as well as addition. Z = {0, −1, +1, −2, +2, −3, +3,...} . The new set Z is not closed under division, however. One therefore expands Z to include fractions as well and arrives at the number field Q, the rational numbers. More formally, the set Q of rational numbers is the set of all ratios of integers: p Q = | p, q ∈ Z ,q=0 q The rational numbers appear to be a very satisfactory algebraic system until one begins to tries to solve equations like x2 =2 . It turns out that there is no rational number that satisfies this equation. To see this, suppose there exists integers p, q such that p 2 2= . q p We can without loss of generality assume that p and q have no common divisors (i.e., that the fraction q is reduced as far as possible). We have 2q2 = p2 so p2 is even. Hence p is even. Therefore, p is of the form p =2k for some k ∈ Z.Butthen 2q2 =4k2 or q2 =2k2 so q is even, so p and q have a common divisor - a contradiction since p and q are can be assumed to be relatively prime. -
Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension fields Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number fields (sometimes number fields for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coefficients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coefficients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by finitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the field of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the field of rational functions in the αi, with rational coefficients. It is the smallest field containing the rational numbers Q and all of the αi. It can be thought of as the field of all expressions built up from elements of Z and the numbers αi by finitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coefficients ai. -
The Evolution of Number in Mathematics Assad Ebrahim
THE EVOLUTION OF NUMBER IN MATHEMATICS ASSAD EBRAHIM Abstract. The number concept has evolved in a gradual process over ten millenia from concrete symbol association in the Middle East c.8000 BCE (“4 sheep-tokens are different than 4 grain-tokens”)1 to an hierarchy of number sets (N ⊂ Z ⊂ Q ⊂ R ⊂ C ⊂ H ⊂ O)2 the last four of which unify number with space and motion.3 In this paper we trace this evolution from whole numbers at the dawn of accounting with number-tokens through the paradoxes of algebraic irrationals, non-solvability by radicals4 , and uncountably many transcendentals, to the physical reality of imaginary numbers and the connection of the octonions with supersymmetry and string theory.5(FAQ3) Our story is of conceptual abstraction, algebraic extension, and analytic completeness, and reflects the modern transformation of mathematics itself. The main paper is brief (four pages) followed by supplements containing FAQs, historical notes, and appendices (holding theorems and proofs), and an annotated bibliography. A project supplement is in preparation to allow learning the material through self-discovery and guided exploration. For scholars and laymen alike it is not philosophy but active experience in mathematics itself that can alone answer the question: ‘What is Mathematics?’ Richard Courant (1941), book of the same title “One of the disappointments experienced by most mathematics students is that they never get a [unified] course on mathematics. They get courses in calculus, algebra, topology, and so on, but the division of labor in teaching seems to prevent these different topics from being combined into a whole. -
Many More Names of (7, 3, 1)
VOL. 88, NO. 2, APRIL 2015 103 Many More Names of (7, 3, 1) EZRA BROWN Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0123 [email protected] Combinatorial designs are collections of subsets of a finite set that satisfy specified conditions, usually involving regularity or symmetry. As the scope of the 984-page Handbook of Combinatorial Designs [7] suggests, this field of study is vast and far reaching. Here is a picture of the very first design to appear in “Opening the Door,” the first of the Handbook’s 109 chapters: Figure 1 The design that opens the door This design, which we call the (7, 3, 1) design, makes appearances in many areas of mathematics. It seems to turn up again and again in unexpected places. An earlier paper in this MAGAZINE [4] described (7, 3, 1)’s appearance in a number of different areas, including finite projective planes, as the Fano plane (FIGURE 1); graph theory, as the Heawood graph and the doubly regular round-robin tournament of order 7; topology, as an arrangement of six mutually adjacent hexagons on the torus; (−1, 1) matrices, as a skew-Hadamard matrix of order 8; and algebraic number theory, as the splitting field of the polynomial (x 2 − 2)(x 2 − 3)(x 2 − 5). In this paper, we show how (7, 3, 1) makes appearances in three areas, namely (1) Hamming’s error-correcting codes, (2) Singer designs and difference sets based on n-dimensional finite projective geometries, and (3) normed algebras. We begin with an overview of block designs, including two descriptions of (7, 3, 1). -
Common and Uncommon Standard Number Sets
Common and Uncommon Standard Number Sets W. Blaine Dowler July 8, 2010 Abstract There are a number of important (and interesting unimportant) sets in mathematics. Sixteen of those sets are detailed here. Contents 1 Natural Numbers 2 2 Whole Numbers 2 3 Prime Numbers 3 4 Composite Numbers 3 5 Perfect Numbers 3 6 Integers 4 7 Rational Numbers 4 8 Algebraic Numbers 5 9 Real Numbers 5 10 Irrational Numbers 6 11 Transcendental Numbers 6 12 Normal Numbers 6 13 Schizophrenic Numbers 7 14 Imaginary Numbers 7 15 Complex Numbers 8 16 Quaternions 8 1 1 Natural Numbers The first set of numbers to be defined is the set of natural numbers. The set is usually labeled N. This set is the smallest set that satisfies the following two conditions: 1. 1 is a natural number, usually written compactly as 1 2 N. 2. If n 2 N, then n + 1 2 N. As this is the smallest set that satisfies these conditions, it is constructed by starting with 1, adding 1 to 1, adding 1 to that, and so forth, such that N = f1; 2; 3; 4;:::g Note that set theorists will often include 0 in the natural numbers. The set of natural numbers was defined formally starting with 1 long before set theorists developed a rigorous way to define numbers as sets. In that formalism, it makes more sense to start with 0, but 1 is the more common standard because it long predates modern set theory. More advanced mathematicians may have encountered \proof by induction." This is a method of completing proofs. -
On Transcendental Numbers: New Results and a Little History
Axioms 2013, xx, 1-x; doi:10.3390/—— OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication On transcendental numbers: new results and a little history Solomon Marcus1 and Florin F. Nichita1,* 1 Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania * Author to whom correspondence should be addressed; E-Mail: [email protected]; Tel: (+40) (0) 21 319 65 06; Fax: (+40) (0) 21 319 65 05 Received: xx / Accepted: xx / Published: xx Abstract: Attempting to create a general framework for studying new results on transcendental numbers, this paper begins with a survey on transcendental numbers and transcendence, it then presents several properties of the transcendental numbers e and π, and then it gives the proofs of new inequalities and identities for transcendental numbers. Also, in relationship with these topics, we study some implications for the theory of the Yang-Baxter equations, and we propose some open problems. Keywords: Euler’s relation, transcendental numbers; transcendental operations / functions; transcendence; Yang-Baxter equation Classification: MSC 01A05; 11D09; 11T23; 33B10; 16T25 arXiv:1502.05637v2 [math.HO] 5 Dec 2017 1. Introduction One of the most famous formulas in mathematics, the Euler’s relation: eπi +1=0 , (1) contains the transcendental numbers e and π, the imaginary number i, the constants 0 and 1, and (transcendental) operations. Beautiful, powerful and surprising, it has changed the mathematics forever. We will unveil profound aspects related to it, and we will propose a counterpart: ei π < e , (2) | − | Axioms 2013, xx 2 Figure 1. An interpretation of the inequality ei π < e . -
Algebraic Number Theory Summary of Notes
Algebraic Number Theory summary of notes Robin Chapman 3 May 2000, revised 28 March 2004, corrected 4 January 2005 This is a summary of the 1999–2000 course on algebraic number the- ory. Proofs will generally be sketched rather than presented in detail. Also, examples will be very thin on the ground. I first learnt algebraic number theory from Stewart and Tall’s textbook Algebraic Number Theory (Chapman & Hall, 1979) (latest edition retitled Algebraic Number Theory and Fermat’s Last Theorem (A. K. Peters, 2002)) and these notes owe much to this book. I am indebted to Artur Costa Steiner for pointing out an error in an earlier version. 1 Algebraic numbers and integers We say that α ∈ C is an algebraic number if f(α) = 0 for some monic polynomial f ∈ Q[X]. We say that β ∈ C is an algebraic integer if g(α) = 0 for some monic polynomial g ∈ Z[X]. We let A and B denote the sets of algebraic numbers and algebraic integers respectively. Clearly B ⊆ A, Z ⊆ B and Q ⊆ A. Lemma 1.1 Let α ∈ A. Then there is β ∈ B and a nonzero m ∈ Z with α = β/m. Proof There is a monic polynomial f ∈ Q[X] with f(α) = 0. Let m be the product of the denominators of the coefficients of f. Then g = mf ∈ Z[X]. Pn j Write g = j=0 ajX . Then an = m. Now n n−1 X n−1+j j h(X) = m g(X/m) = m ajX j=0 1 is monic with integer coefficients (the only slightly problematical coefficient n −1 n−1 is that of X which equals m Am = 1). -
Approximation to Real Numbers by Algebraic Numbers of Bounded Degree
Approximation to real numbers by algebraic numbers of bounded degree (Review of existing results) Vladislav Frank University Bordeaux 1 ALGANT Master program May,2007 Contents 1 Approximation by rational numbers. 2 2 Wirsing conjecture and Wirsing theorem 4 3 Mahler and Koksma functions and original Wirsing idea 6 4 Davenport-Schmidt method for the case d = 2 8 5 Linear forms and the subspace theorem 10 6 Hopeless approach and Schmidt counterexample 12 7 Modern approach to Wirsing conjecture 12 8 Integral approximation 18 9 Extremal numbers due to Damien Roy 22 10 Exactness of Schmidt result 27 1 1 Approximation by rational numbers. It seems, that the problem of approximation of given number by numbers of given class was firstly stated by Dirichlet. So, we may call his theorem as ”the beginning of diophantine approximation”. Theorem 1.1. (Dirichlet, 1842) For every irrational number ζ there are in- p finetely many rational numbers q , such that p 1 0 < ζ − < . q q2 Proof. Take a natural number N and consider numbers {qζ} for all q, 1 ≤ q ≤ N. They all are in the interval (0, 1), hence, there are two of them with distance 1 not exceeding q . Denote the corresponding q’s as q1 and q2. So, we know, that 1 there are integers p1, p2 ≤ N such that |(q2ζ − p2) − (q1ζ − p1)| < N . Hence, 1 for q = q2 − q1 and p = p2 − p1 we have |qζ − p| < . Division by q gives N ζ − p < 1 ≤ 1 . So, for every N we have an approximation with precision q qN q2 1 1 qN < N . -
CALCULUS for the UTTERLY CONFUSED Has Proven to Be a Wonderful Review Enabling Me to Move Forward in Application of Calculus and Advanced Topics in Mathematics
TLFeBOOK WHAT READERS ARE SAYIN6 "I wish I had had this book when I needed it most, which was during my pre-med classes. It could have also been a great tool for me in a few medical school courses." Or. Kellie Aosley8 Recent Hedical school &a&ate "CALCULUS FOR THE UTTERLY CONFUSED has proven to be a wonderful review enabling me to move forward in application of calculus and advanced topics in mathematics. I found it easy to use and great as a reference for those darker aspects of calculus. I' Aaron Ladeville, Ekyiheeriky Student 'I1 am so thankful for CALCULUS FOR THE UTTERLY CONFUSED! I started out Clueless but ended with an All' Erika Dickstein8 0usihess school Student "As a non-traditional student one thing I have learned is the importance of material supplementary to texts. Especially in calculus it helps to have a second source, especially one as lucid and fun to read as CALCULUS FOR THE UTTERtY CONFUSED. Anyone, whether you are a math weenie or not, will get something out of this book. With this book, your chances of survival in the calculus jungle are greatly increased.'I Brad &3~ker,Physics Student Other books in the Utterly Conhrsed Series include: Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly Confrred CALCULUS FOR THE UTTERLY CONFUSED Robert M. Oman Daniel M. Oman McGraw -Hill New York San Francisco Washington, D.C. Auckland Bogoth Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto Library of Congress Cataloging-in-Publication Data Oman, Robert M. -
Transcendental Numbers
INTRODUCTION TO TRANSCENDENTAL NUMBERS VO THANH HUAN Abstract. The study of transcendental numbers has developed into an enriching theory and constitutes an important part of mathematics. This report aims to give a quick overview about the theory of transcen- dental numbers and some of its recent developments. The main focus is on the proof that e is transcendental. The Hilbert's seventh problem will also be introduced. 1. Introduction Transcendental number theory is a branch of number theory that concerns about the transcendence and algebraicity of numbers. Dated back to the time of Euler or even earlier, it has developed into an enriching theory with many applications in mathematics, especially in the area of Diophantine equations. Whether there is any transcendental number is not an easy question to answer. The discovery of the first transcendental number by Liouville in 1851 sparked up an interest in the field and began a new era in the theory of transcendental number. In 1873, Charles Hermite succeeded in proving that e is transcendental. And within a decade, Lindemann established the tran- scendence of π in 1882, which led to the impossibility of the ancient Greek problem of squaring the circle. The theory has progressed significantly in recent years, with answer to the Hilbert's seventh problem and the discov- ery of a nontrivial lower bound for linear forms of logarithms of algebraic numbers. Although in 1874, the work of Georg Cantor demonstrated the ubiquity of transcendental numbers (which is quite surprising), finding one or proving existing numbers are transcendental may be extremely hard. In this report, we will focus on the proof that e is transcendental. -
Arxiv:1103.4922V1 [Math.NT] 25 Mar 2011 Hoyo Udai Om.Let Forms
QUATERNION ORDERS AND TERNARY QUADRATIC FORMS STEFAN LEMURELL Introduction The main purpose of this paper is to provide an introduction to the arith- metic theory of quaternion algebras. However, it also contains some new results, most notably in Section 5. We will emphasise on the connection between quaternion algebras and quadratic forms. This connection will pro- vide us with an efficient tool to consider arbitrary orders instead of having to restrict to special classes of them. The existing results are mostly restricted to special classes of orders, most notably to so called Eichler orders. The paper is organised as follows. Some notations and background are provided in Section 1, especially on the theory of quadratic forms. Section 2 contains the basic theory of quaternion algebras. Moreover at the end of that section, we give a quite general solution to the problem of representing a quaternion algebra with given discriminant. Such a general description seems to be lacking in the literature. Section 3 gives the basic definitions concerning orders in quaternion alge- bras. In Section 4, we prove an important correspondence between ternary quadratic forms and quaternion orders. Section 5 deals with orders in quaternion algebras over p-adic fields. The major part is an investigation of the isomorphism classes in the non-dyadic and 2-adic cases. The starting- point is the correspondence with ternary quadratic forms and known classi- fications of such forms. From this, we derive representatives of the isomor- phism classes of quaternion orders. These new results are complements to arXiv:1103.4922v1 [math.NT] 25 Mar 2011 existing more ring-theoretic descriptions of orders.