Two Variations of a Theorem of Kronecker
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Algebraic Generality Vs Arithmetic Generality in the Controversy Between C
Algebraic generality vs arithmetic generality in the controversy between C. Jordan and L. Kronecker (1874). Frédéric Brechenmacher (1). Introduction. Throughout the whole year of 1874, Camille Jordan and Leopold Kronecker were quarrelling over two theorems. On the one hand, Jordan had stated in his 1870 Traité des substitutions et des équations algébriques a canonical form theorem for substitutions of linear groups; on the other hand, Karl Weierstrass had introduced in 1868 the elementary divisors of non singular pairs of bilinear forms (P,Q) in stating a complete set of polynomial invariants computed from the determinant |P+sQ| as a key theorem of the theory of bilinear and quadratic forms. Although they would be considered equivalent as regard to modern mathematics (2), not only had these two theorems been stated independently and for different purposes, they had also been lying within the distinct frameworks of two theories until some connections came to light in 1872-1873, breeding the 1874 quarrel and hence revealing an opposition over two practices relating to distinctive cultural features. As we will be focusing in this paper on how the 1874 controversy about Jordan’s algebraic practice of canonical reduction and Kronecker’s arithmetic practice of invariant computation sheds some light on two conflicting perspectives on polynomial generality we shall appeal to former publications which have already been dealing with some of the cultural issues highlighted by this controversy such as tacit knowledge, local ways of thinking, internal philosophies and disciplinary ideals peculiar to individuals or communities [Brechenmacher 200?a] as well as the different perceptions expressed by the two opponents of a long term history involving authors such as Joseph-Louis Lagrange, Augustin Cauchy and Charles Hermite [Brechenmacher 200?b]. -
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. -
On the Rational Approximations to the Powers of an Algebraic Number: Solution of Two Problems of Mahler and Mend S France
Acta Math., 193 (2004), 175 191 (~) 2004 by Institut Mittag-Leffier. All rights reserved On the rational approximations to the powers of an algebraic number: Solution of two problems of Mahler and Mend s France by PIETRO CORVAJA and UMBERTO ZANNIER Universith di Udine Scuola Normale Superiore Udine, Italy Pisa, Italy 1. Introduction About fifty years ago Mahler [Ma] proved that if ~> 1 is rational but not an integer and if 0<l<l, then the fractional part of (~n is larger than l n except for a finite set of integers n depending on ~ and I. His proof used a p-adic version of Roth's theorem, as in previous work by Mahler and especially by Ridout. At the end of that paper Mahler pointed out that the conclusion does not hold if c~ is a suitable algebraic number, as e.g. 1 (1 + x/~ ) ; of course, a counterexample is provided by any Pisot number, i.e. a real algebraic integer c~>l all of whose conjugates different from cr have absolute value less than 1 (note that rational integers larger than 1 are Pisot numbers according to our definition). Mahler also added that "It would be of some interest to know which algebraic numbers have the same property as [the rationals in the theorem]". Now, it seems that even replacing Ridout's theorem with the modern versions of Roth's theorem, valid for several valuations and approximations in any given number field, the method of Mahler does not lead to a complete solution to his question. One of the objects of the present paper is to answer Mahler's question completely; our methods will involve a suitable version of the Schmidt subspace theorem, which may be considered as a multi-dimensional extension of the results mentioned by Roth, Mahler and Ridout. -
The Science of Infinity
Volume 22, Number 2, April/May 2016 The Science of Infinity Raphael A. Fraser, PhD, Division of Biostatistics, MCW \The infinite! No other question has ever moved so profoundly the spirit of man," said David Hilbert (1862-1943), one of the most influential mathematicians of the 19th century. The subject has been studied extensively by mathematicians and philosophers but still remains an enigma of the intellectual world. The notions of “finite” and “infinite” are primitive, and it is very likely that the reader has never examined these notions very carefully. For example, there are as many even numbers as there are natural numbers. This is true but counter intuitive. We may argue, that the set of even numbers is half the size of the set of natural numbers, thus how can both sets be of the same size? To answer this question we must first explain what do we mean when we say two sets are the same size? Normally, we would count the elements in each set then verify the counts are the same. However, there is another way to verify two sets are the same size by matching. To illustrate, that matching is more fundamental than counting let us consider an example. If I walked into a large room with many chairs but all the seats are occupied and no one is standing then I can conclude that there are as many people as there are chairs. That is, there is a one-to-one correspondence between person and chair. Hence both sets are the same size, even though we did not count how many persons or chairs they were. -
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 . -
Mop 2018: Algebraic Conjugates and Number Theory (06/15, Bk)
MOP 2018: ALGEBRAIC CONJUGATES AND NUMBER THEORY (06/15, BK) VICTOR WANG 1. Arithmetic properties of polynomials Problem 1.1. If P; Q 2 Z[x] share no (complex) roots, show that there exists a finite set of primes S such that p - gcd(P (n);Q(n)) holds for all primes p2 = S and integers n 2 Z. Problem 1.2. If P; Q 2 C[t; x] share no common factors, show that there exists a finite set S ⊂ C such that (P (t; z);Q(t; z)) 6= (0; 0) holds for all complex numbers t2 = S and z 2 C. Problem 1.3 (USA TST 2010/1). Let P 2 Z[x] be such that P (0) = 0 and gcd(P (0);P (1);P (2);::: ) = 1: Show there are infinitely many n such that gcd(P (n) − P (0);P (n + 1) − P (1);P (n + 2) − P (2);::: ) = n: Problem 1.4 (Calvin Deng). Is R[x]=(x2 + 1)2 isomorphic to C[y]=y2 as an R-algebra? Problem 1.5 (ELMO 2013, Andre Arslan, one-dimensional version). For what polynomials P 2 Z[x] can a positive integer be assigned to every integer so that for every integer n ≥ 1, the sum of the n1 integers assigned to any n consecutive integers is divisible by P (n)? 2. Algebraic conjugates and symmetry 2.1. General theory. Let Q denote the set of algebraic numbers (over Q), i.e. roots of polynomials in Q[x]. Proposition-Definition 2.1. If α 2 Q, then there is a unique monic polynomial M 2 Q[x] of lowest degree with M(α) = 0, called the minimal polynomial of α over Q. -
Algebraic Numbers, Algebraic Integers
Algebraic Numbers, Algebraic Integers October 7, 2012 1 Reading Assignment: 1. Read [I-R]Chapter 13, section 1. 2. Suggested reading: Davenport IV sections 1,2. 2 Homework set due October 11 1. Let p be a prime number. Show that the polynomial f(X) = Xp−1 + Xp−2 + Xp−3 + ::: + 1 is irreducible over Q. Hint: Use Eisenstein's Criterion that you have proved in the Homework set due today; and find a change of variables, making just the right substitution for X to turn f(X) into a polynomial for which you can apply Eisenstein' criterion. 2. Show that any unit not equal to ±1 in the ring of integers of a real quadratic field is of infinite order in the group of units of that ring. 3. [I-R] Page 201, Exercises 4,5,7,10. 4. Solve Problems 1,2 in these notes. 3 Recall: Algebraic integers; rings of algebraic integers Definition 1 An algebraic integer is a root of a monic polynomial with rational integer coefficients. Proposition 1 Let θ 2 C be an algebraic integer. The minimal1 monic polynomial f(X) 2 Q[X] having θ as a root, has integral coefficients; that is, f(X) 2 Z[X] ⊂ Q[X]. 1i.e., minimal degree 1 Proof: This follows from the fact that a product of primitive polynomials is again primitive. Discuss. Define content. Note that this means that there is no ambiguity in the meaning of, say, quadratic algebraic integer. It means, equivalently, a quadratic number that is an algebraic integer, or number that satisfies a quadratic monic polynomial relation with integral coefficients, or: Corollary 1 A quadratic number is a quadratic integer if and only if its trace and norm are integers. -
Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial
Rose-Hulman Undergraduate Mathematics Journal Volume 22 Issue 1 Article 1 Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial Richard B. Schneider University of Missouri - Kansas City, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Part of the Analysis Commons, European History Commons, and the Intellectual History Commons Recommended Citation Schneider, Richard B. (2021) "Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial," Rose-Hulman Undergraduate Mathematics Journal: Vol. 22 : Iss. 1 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/1 Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial Cover Page Footnote The author would like to thank Dr. Richard Delaware for his guidance and support throughout the completion of this work. This article is available in Rose-Hulman Undergraduate Mathematics Journal: https://scholar.rose-hulman.edu/rhumj/ vol22/iss1/1 Rose-Hulman Undergraduate Mathematics Journal VOLUME 22, ISSUE 1, 2021 Irrational Philosophy? Kronecker’s Constructive Philosophy and Finding the Real Roots of a Polynomial By Richard B. Schneider Abstract. The prominent mathematician Leopold Kronecker (1823 – 1891) is often rel- egated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. -
Kbonecker and His Arithmetical Theory of the Algebraic Equation
THE ALGEBRAIC EQUATION. 173 KBONECKER AND HIS ARITHMETICAL THEORY OF THE ALGEBRAIC EQUATION. LEOPOLD KRONECKER, one of the most illustrious of con temporary mathematicians, died at Berlin on the 29th of last December in his 68th year. For many years he had been one of the famous mathe maticians of Germany and at the time of his death was senior active professor of the mathematical faculty of the University of Berlin and editor in chief of the Journal für reine und angewandte Mathematik (Orelle). He was the last of the great triumvirate—Kummer, Weier- strass, Kronecker—to be lost to the university. Kummer retired nearly ten years ago because of sickness and old age, and recently Weierstrass followed him ; but younger than the other two, Kronecker was overtaken by death in the midst of the work to which his life had been devoted. Despite his years he was much too early lost to science. The genius which had enriched mathematical literature with so many profound and beautiful researches showed no signs of weak ness or weariness. Kronecker was born at Liegnitz near Breslau in 1823. While yet a boy at the Gymnasium of his native town his fine mathematical talents attracted the notice of his master, Kummer, whose distinguished career was then just beginning. Kummer's persuasions rescued him from the business career for which he was preparing and brought him to the univer sity.* He studied at Breslau, whither in the meantime Kummer had been called, Bonn, and Berlin, making his degree at Berlin in 1845 with a dissertation of great value : De unitati- bus complezis. -
Totally Ramified Primes and Eisenstein Polynomials
TOTALLY RAMIFIED PRIMES AND EISENSTEIN POLYNOMIALS KEITH CONRAD 1. Introduction A (monic) polynomial in Z[T ], n n−1 f(T ) = T + cn−1T + ··· + c1T + c0; is Eisenstein at a prime p when each coefficient ci is divisible by p and the constant term c0 is not divisible by p2. Such polynomials are irreducible in Q[T ], and this Eisenstein criterion for irreducibility is the way essentially everyone first meets Eisenstein polynomials. Here we will show Eisenstein polynomials at a prime p give us useful information about p-divisibility of coefficients of a power basis and ramification of p in a number field. Let K be a number field, with degree n over Q. A primep number p is said to be totally n 2 ramified in Kpwhen pOK = p . For example, in Z[i] and Z[ p−5] we have (2) = (1 + i) and (2) = (2; 1 + −5)2, so 2 is totally ramified in Q(i) and Q( −5). 2. Eisenstein polynomials and p-divisibility of coefficients Our first use of Eisenstein polynomials will be to extract information about coefficients for algebraic integers in the power basis generated by the root of an Eisenstein polynomial. Lemma 2.1. Let K=Q be a number field with degree n. Assume K = Q(α), where α 2 OK and its minimal polynomial over Q is Eisenstein at p. For a0; a1; : : : ; an−1 2 Z, if n−1 (2.1) a0 + a1α + ··· + an−1α ≡ 0 mod pOK ; then ai ≡ 0 mod pZ for all i. Proof. Assume for some j 2 f0; 1; : : : ; n − 1g that ai ≡ 0 mod pZ for i < j (this is an empty condition for j = 0). -
Chapter 3 Algebraic Numbers and Algebraic Number Fields
Chapter 3 Algebraic numbers and algebraic number fields Literature: S. Lang, Algebra, 2nd ed. Addison-Wesley, 1984. Chaps. III,V,VII,VIII,IX. P. Stevenhagen, Dictaat Algebra 2, Algebra 3 (Dutch). We have collected some facts about algebraic numbers and algebraic number fields that are needed in this course. Many of the results are stated without proof. For proofs and further background, we refer to Lang's book mentioned above, Peter Stevenhagen's Dutch lecture notes on algebra, and any basic text book on algebraic number theory. We do not require any pre-knowledge beyond basic ring theory. In the Appendix (Section 3.4) we have included some general theory on ring extensions for the interested reader. 3.1 Algebraic numbers and algebraic integers A number α 2 C is called algebraic if there is a non-zero polynomial f 2 Q[X] with f(α) = 0. Otherwise, α is called transcendental. We define the algebraic closure of Q by Q := fα 2 C : α algebraicg: Lemma 3.1. (i) Q is a subfield of C, i.e., sums, differences, products and quotients of algebraic numbers are again algebraic; 35 n n−1 (ii) Q is algebraically closed, i.e., if g = X + β1X ··· + βn 2 Q[X] and α 2 C is a zero of g, then α 2 Q. (iii) If g 2 Q[X] is a monic polynomial, then g = (X − α1) ··· (X − αn) with α1; : : : ; αn 2 Q. Proof. This follows from some results in the Appendix (Section 3.4). Proposition 3.24 in Section 3.4 with A = Q, B = C implies that Q is a ring. -
An Exposition of the Eisenstein Integers
Eastern Illinois University The Keep Masters Theses Student Theses & Publications 2016 An Exposition of the Eisenstein Integers Sarada Bandara Eastern Illinois University This research is a product of the graduate program in Mathematics and Computer Science at Eastern Illinois University. Find out more about the program. Recommended Citation Bandara, Sarada, "An Exposition of the Eisenstein Integers" (2016). Masters Theses. 2467. https://thekeep.eiu.edu/theses/2467 This is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Masters Theses by an authorized administrator of The Keep. For more information, please contact [email protected]. The Graduate School� EASTERNILLINOIS UNIVERSITY' Thesis Maintenance and Reproduction Certificate FOR: Graduate Candidates Completing Theses in Partial Fulfillment of the Degree Graduate Faculty Advisors Directing the Theses Preservation, Reproduction, and Distribution of Thesis Research RE: Preserving, reproducing, and distributing thesis research is an important part of Booth Library's responsibility to provide access to scholarship. In order to further this goal, Booth Library makes all graduate theses completed as part of a degree program at Eastern Illinois University available for personal study, research, and other not-for-profit educational purposes. Under 17 U.S.C. § 108, the library may reproduce and distribute a copy without infringing on copyright; however, professional courtesy dictates that permission be requested from the author before doing so. Your signatures affirm the following: • The graduate candidate is the author of this thesis. • The graduate candidate retains the copyright and intellectual property rights associated with the original research, creative activity, and intellectual or artistic content of the thesis.