The General Quintic Equation, Its Solution by Factorization Into Cubic
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Arxiv:1406.1948V2 [Math.GM] 18 Dec 2014 Hr = ∆ Where H Ouino 1 Is (1) of Solution the 1
Solution of Polynomial Equations with Nested Radicals Nikos Bagis Stenimahou 5 Edessa Pellas 58200, Greece [email protected] Abstract In this article we present solutions of arbitrary polynomial equations in nested periodic radicals Keywords: Nested Radicals; Quintic; Sextic; Septic; Equations; Polynomi- als; Solutions; Higher functions; 1 A general type of equations Consider the following general type of equations ax2µ + bxµ + c = xν . (1) Then (1) can be written in the form b 2 ∆2 a xµ + = xν , (2) 2a − 4a where ∆ = √b2 4ac. Hence − 2 µ b ∆ 1 ν x = − + 2 + x (3) 2a r4a a Set now √d x = x1/d, d Q 0 , then ∈ + −{ } 2 µ b ∆ 1 x = − + + xν (4) s 2a 4a2 a arXiv:1406.1948v2 [math.GM] 18 Dec 2014 r hence µ/ν 2 ν b ∆ 1 ν x = − + 2 + x (5) s 2a r4a a Theorem 1. The solution of (1) is µ v 2 µ/ν 2 2 u b v ∆ 1 b ∆ 1 µ/ν b ∆ x = u− + u + v− + v + − + + ... (6) u 2a u4a2 a u 2a u4a2 a s 2a 4a2 u u u u r u u u t t t t 1 Proof. Use relation (4) and repeat it infinite times. Corollary 1. The equation ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0 (7) is always solvable with nested radicals. Proof. We know (see [1]) that all sextic equations, of the most general form (7) are equivalent by means of a Tchirnhausen transform y = kx4 + lx3 + mx2 + nx + s (8) to the form 6 2 y + e1y + f1y + g1 = 0 (9) But the last equation is of the form (1) with µ = 1 and ν = 6 and have solution 2 1/6 2 2 f v ∆ 1 f ∆ 1 1/6 f ∆ y = 1 + u 1 v 1 + 1 1 + 1 + .. -
1. Make Sense of Problems and Persevere in Solving Them. Mathematically Proficient Students Start by Explaining to Themselves T
1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. -
ALGEBRAIC GEOMETRY (1) Compute the Points of Intersection
ALGEBRAIC GEOMETRY HOMEWORK 5 (1) Compute the points of intersection of the circle x2 +y2 +x−y = 0 with the lemniscate (x2 +y2)2 = x2 −y2 using resultants (can you compute the affine points of intersections directly?), and compute the multiplicities. Explain which coordinate system you choose, how the equations look like, what the resultant is, and what its factors are. (2) Here you will learn how to solve cubics using resultants. Assume you want to solve the cubic equation x3 + ax2 + bx + c = 0. (a) Show that the substitution y = x + a/3 transforms the cubic into y3 + By + C = 0. (b) Now assume that we have to solve x3 + bx + c = 0. Using linear re- lations y = rx ∗ b will not make the linear term disappear (without bringing back the quadratic term). Thus we try to put y = x2 + cx + d and then eliminate x using resultants: type in p = polresultant(x^3+b*x+c,y-x^2-d*x-e,x) polcoeff(p,2,y) polcoeff(p,1,y) (you can find out what polcoeff is doing by typing ?polcoeff). Now we know that we can keep the quadratic term out of the equation by choosing e = 2b/3. With this value of e go back and recompute the resultant; this will now be a cubic in y without quadratic term. The coefficient of the linear term can be made to disappear by solving a quadratic equation (which is called the resolvent of the original cubic). (c) Thus you can solve the cubic by solving the resolvent, using the trans- formations above to turn the cubic into a pure cubic y3 + c, which in turn is easy to solve. -
Higher Mathematics
; HIGHER MATHEMATICS Chapter I. THE SOLUTION OF EQUATIONS. By Mansfield Merriman, Professor of Civil Engineering in Lehigh University. Art. 1. Introduction. In this Chapter will be presented a brief outline of methods, not commonly found in text-books, for the solution of an equation containing one unknown quantity. Graphic, numeric, and algebraic solutions will be given by which the real roots of both algebraic and transcendental equations may be ob- tained, together with historical information and theoretic discussions. An algebraic equation is one that involves only the opera- tions of arithmetic. It is to be first freed from radicals so as to make the exponents of the unknown quantity all integers the degree of the equation is then indicated by the highest ex- ponent of the unknown quantity. The algebraic solution of an algebraic equation is the expression of its roots in terms of literal is the coefficients ; this possible, in general, only for linear, quadratic, cubic, and quartic equations, that is, for equations of the first, second, third, and fourth degrees. A numerical equation is an algebraic equation having all its coefficients real numbers, either positive or negative. For the four degrees 2 THE SOLUTION OF EQUATIONS. [CHAP. I. above mentioned the roots of numerical equations may be computed from the formulas for the algebraic solutions, unless they fall under the so-called irreducible case wherein real quantities are expressed in imaginary forms. An algebraic equation of the n th degree may be written with all its terms transposed to the first member, thus: n- 1 2 x" a x a,x"- . -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Some Properties of the Discriminant Matrices of a Linear Associative Algebra*
570 R. F. RINEHART [August, SOME PROPERTIES OF THE DISCRIMINANT MATRICES OF A LINEAR ASSOCIATIVE ALGEBRA* BY R. F. RINEHART 1. Introduction. Let A be a linear associative algebra over an algebraic field. Let d, e2, • • • , en be a basis for A and let £»•/*., (hjik = l,2, • • • , n), be the constants of multiplication corre sponding to this basis. The first and second discriminant mat rices of A, relative to this basis, are defined by Ti(A) = \\h(eres[ CrsiCij i, j=l T2(A) = \\h{eres / ,J CrsiC j II i,j=l where ti(eres) and fa{erea) are the first and second traces, respec tively, of eres. The first forms in terms of the constants of multi plication arise from the isomorphism between the first and sec ond matrices of the elements of A and the elements themselves. The second forms result from direct calculation of the traces of R(er)R(es) and S(er)S(es), R{ei) and S(ei) denoting, respectively, the first and second matrices of ei. The last forms of the dis criminant matrices show that each is symmetric. E. Noetherf and C. C. MacDuffeeJ discovered some of the interesting properties of these matrices, and shed new light on the particular case of the discriminant matrix of an algebraic equation. It is the purpose of this paper to develop additional properties of these matrices, and to interpret them in some fa miliar instances. Let A be subjected to a transformation of basis, of matrix M, 7 J rH%j€j 1,2, *0). -
Section X.56. Insolvability of the Quintic
X.56 Insolvability of the Quintic 1 Section X.56. Insolvability of the Quintic Note. Now is a good time to reread the first set of notes “Why the Hell Am I in This Class?” As we have claimed, there is the quadratic formula to solve all polynomial equations ax2 +bx+c = 0 (in C, say), there is a cubic equation to solve ax3+bx2+cx+d = 0, and there is a quartic equation to solve ax4+bx3+cx2+dx+e = 0. However, there is not a general algebraic equation which solves the quintic ax5 + bx4 + cx3 + dx2 + ex + f = 0. We now have the equipment to establish this “insolvability of the quintic,” as well as a way to classify which polynomial equations can be solved algebraically (that is, using a finite sequence of operations of addition [or subtraction], multiplication [or division], and taking of roots [or raising to whole number powers]) in a field F . Definition 56.1. An extension field K of a field F is an extension of F by radicals if there are elements α1, α2,...,αr ∈ K and positive integers n1,n2,...,nr such that n1 ni K = F (α1, α2,...,αr), where α1 ∈ F and αi ∈ F (α1, α2,...,αi−1) for 1 <i ≤ r. A polynomial f(x) ∈ F [x] is solvable by radicals over F if the splitting field E of f(x) over F is contained in an extension of F by radicals. n1 Note. The idea in this definition is that F (α1) includes the n1th root of α1 ∈ F . -
A Historical Survey of Methods of Solving Cubic Equations Minna Burgess Connor
University of Richmond UR Scholarship Repository Master's Theses Student Research 7-1-1956 A historical survey of methods of solving cubic equations Minna Burgess Connor Follow this and additional works at: http://scholarship.richmond.edu/masters-theses Recommended Citation Connor, Minna Burgess, "A historical survey of methods of solving cubic equations" (1956). Master's Theses. Paper 114. This Thesis is brought to you for free and open access by the Student Research at UR Scholarship Repository. It has been accepted for inclusion in Master's Theses by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. A HISTORICAL SURVEY OF METHODS OF SOLVING CUBIC E<~UATIONS A Thesis Presented' to the Faculty or the Department of Mathematics University of Richmond In Partial Fulfillment ot the Requirements tor the Degree Master of Science by Minna Burgess Connor August 1956 LIBRARY UNIVERStTY OF RICHMOND VIRGlNIA 23173 - . TABLE Olf CONTENTS CHAPTER PAGE OUTLINE OF HISTORY INTRODUCTION' I. THE BABYLONIANS l) II. THE GREEKS 16 III. THE HINDUS 32 IV. THE CHINESE, lAPANESE AND 31 ARABS v. THE RENAISSANCE 47 VI. THE SEVEW.l'EEl'iTH AND S6 EIGHTEENTH CENTURIES VII. THE NINETEENTH AND 70 TWENTIETH C:BNTURIES VIII• CONCLUSION, BIBLIOGRAPHY 76 AND NOTES OUTLINE OF HISTORY OF SOLUTIONS I. The Babylonians (1800 B. c.) Solutions by use ot. :tables II. The Greeks·. cs·oo ·B.c,. - )00 A~D.) Hippocrates of Chios (~440) Hippias ot Elis (•420) (the quadratrix) Archytas (~400) _ .M~naeobmus J ""375) ,{,conic section~) Archimedes (-240) {conioisections) Nicomedea (-180) (the conchoid) Diophantus ot Alexander (75) (right-angled tr~angle) Pappus (300) · III. -
501 Algebra Questions 2Nd Edition
501 Algebra Questions 501 Algebra Questions 2nd Edition ® NEW YORK Copyright © 2006 LearningExpress, LLC. All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by LearningExpress, LLC, New York. Library of Congress Cataloging-in-Publication Data: 501 algebra questions.—2nd ed. p. cm. Rev. ed. of: 501 algebra questions / [William Recco]. 1st ed. © 2002. ISBN 1-57685-552-X 1. Algebra—Problems, exercises, etc. I. Recco, William. 501 algebra questions. II. LearningExpress (Organization). III. Title: Five hundred one algebra questions. IV. Title: Five hundred and one algebra questions. QA157.A15 2006 512—dc22 2006040834 Printed in the United States of America 98765432 1 Second Edition ISBN 1-57685-552-X For more information or to place an order, contact LearningExpress at: 55 Broadway 8th Floor New York, NY 10006 Or visit us at: www.learnatest.com The LearningExpress Skill Builder in Focus Writing Team is comprised of experts in test preparation, as well as educators and teachers who specialize in language arts and math. LearningExpress Skill Builder in Focus Writing Team Brigit Dermott Freelance Writer English Tutor, New York Cares New York, New York Sandy Gade Project Editor LearningExpress New York, New York Kerry McLean Project Editor Math Tutor Shirley, New York William Recco Middle School Math Teacher, Grade 8 New York Shoreham/Wading River School District Math Tutor St. James, New York Colleen Schultz Middle School Math Teacher, Grade 8 Vestal Central School District Math Tutor -
The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work. -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Introduction to Complex Analysis Michael Taylor
Introduction to Complex Analysis Michael Taylor 1 2 Contents Chapter 1. Basic calculus in the complex domain 0. Complex numbers, power series, and exponentials 1. Holomorphic functions, derivatives, and path integrals 2. Holomorphic functions defined by power series 3. Exponential and trigonometric functions: Euler's formula 4. Square roots, logs, and other inverse functions I. π2 is irrational Chapter 2. Going deeper { the Cauchy integral theorem and consequences 5. The Cauchy integral theorem and the Cauchy integral formula 6. The maximum principle, Liouville's theorem, and the fundamental theorem of al- gebra 7. Harmonic functions on planar regions 8. Morera's theorem, the Schwarz reflection principle, and Goursat's theorem 9. Infinite products 10. Uniqueness and analytic continuation 11. Singularities 12. Laurent series C. Green's theorem F. The fundamental theorem of algebra (elementary proof) L. Absolutely convergent series Chapter 3. Fourier analysis and complex function theory 13. Fourier series and the Poisson integral 14. Fourier transforms 15. Laplace transforms and Mellin transforms H. Inner product spaces N. The matrix exponential G. The Weierstrass and Runge approximation theorems Chapter 4. Residue calculus, the argument principle, and two very special functions 16. Residue calculus 17. The argument principle 18. The Gamma function 19. The Riemann zeta function and the prime number theorem J. Euler's constant S. Hadamard's factorization theorem 3 Chapter 5. Conformal maps and geometrical aspects of complex function the- ory 20. Conformal maps 21. Normal families 22. The Riemann sphere (and other Riemann surfaces) 23. The Riemann mapping theorem 24. Boundary behavior of conformal maps 25. Covering maps 26.