Solving Polynomial Equations by Radicals
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Symmetry and the Cubic Formula
SYMMETRY AND THE CUBIC FORMULA DAVID CORWIN The purpose of this talk is to derive the cubic formula. But rather than finding the exact formula, I'm going to prove that there is a cubic formula. The way I'm going to do this uses symmetry in a very elegant way, and it foreshadows Galois theory. Note that this material comes almost entirely from the first chapter of http://homepages.warwick.ac.uk/~masda/MA3D5/Galois.pdf. 1. Deriving the Quadratic Formula Before discussing the cubic formula, I would like to consider the quadratic formula. While I'm expecting you know the quadratic formula already, I'd like to treat this case first in order to motivate what we will do with the cubic formula. Suppose we're solving f(x) = x2 + bx + c = 0: We know this factors as f(x) = (x − α)(x − β) where α; β are some complex numbers, and the problem is to find α; β in terms of b; c. To go the other way around, we can multiply out the expression above to get (x − α)(x − β) = x2 − (α + β)x + αβ; which means that b = −(α + β) and c = αβ. Notice that each of these expressions doesn't change when we interchange α and β. This should be the case, since after all, we labeled the two roots α and β arbitrarily. This means that any expression we get for α should equally be an expression for β. That is, one formula should produce two values. We say there is an ambiguity here; it's ambiguous whether the formula gives us α or β. -
Solving Cubic Polynomials
Solving Cubic Polynomials 1.1 The general solution to the quadratic equation There are four steps to finding the zeroes of a quadratic polynomial. 1. First divide by the leading term, making the polynomial monic. a 2. Then, given x2 + a x + a , substitute x = y − 1 to obtain an equation without the linear term. 1 0 2 (This is the \depressed" equation.) 3. Solve then for y as a square root. (Remember to use both signs of the square root.) a 4. Once this is done, recover x using the fact that x = y − 1 . 2 For example, let's solve 2x2 + 7x − 15 = 0: First, we divide both sides by 2 to create an equation with leading term equal to one: 7 15 x2 + x − = 0: 2 2 a 7 Then replace x by x = y − 1 = y − to obtain: 2 4 169 y2 = 16 Solve for y: 13 13 y = or − 4 4 Then, solving back for x, we have 3 x = or − 5: 2 This method is equivalent to \completing the square" and is the steps taken in developing the much- memorized quadratic formula. For example, if the original equation is our \high school quadratic" ax2 + bx + c = 0 then the first step creates the equation b c x2 + x + = 0: a a b We then write x = y − and obtain, after simplifying, 2a b2 − 4ac y2 − = 0 4a2 so that p b2 − 4ac y = ± 2a and so p b b2 − 4ac x = − ± : 2a 2a 1 The solutions to this quadratic depend heavily on the value of b2 − 4ac. -
F[X] Be an Irreducible Cubic X 3 + Ax 2 + Bx + Cw
Math 404 Assignment 3. Due Friday, May 3, 2013. Cubic equations. Let f(x) F [x] be an irreducible cubic x3 + ax2 + bx + c with roots ∈ x1, x2, x3, and splitting field K/F . Since each element of the Galois group permutes the roots, G(K/F ) is a subgroup of S3, the group of permutations of the three roots, and [K : F ] = G(K/F ) divides (S3) = 6. Since [F (x1): F ] = 3, we see that [K : F ] equals ◦ ◦ 3 or 6, and G(K/F ) is either A3 or S3. If G(K/F ) = S3, then K has a unique subfield of dimension 2, namely, KA3 . We have seen that the determinant J of the Jacobian matrix of the partial derivatives of the system a = (x1 + x2 + x3) − b = x1x2 + x2x3 + x3x1 c = (x1x2x3) − equals (x1 x2)(x2 x3)(x1 x3). − − − Formula. J 2 = a2b2 4a3c 4b3 27c2 + 18abc F . − − − ∈ An odd permutation of the roots takes J =(x1 x2)(x2 x3)(x1 x3) to J and an even permutation of the roots takes J to J. − − − − 1. Let f(x) F [x] be an irreducible cubic polynomial. ∈ (a). Show that, if J is an element of K, then the Galois group G(L/K) is the alternating group A3. Solution. If J F , then every element of G(K/F ) fixes J, and G(K/F ) must be A3, ∈ (b). Show that, if J is not an element of F , then the splitting field K of f(x) F [x] has ∈ Galois group G(K/F ) isomorphic to S3. -
January 10, 2010 CHAPTER SIX IRREDUCIBILITY and FACTORIZATION §1. BASIC DIVISIBILITY THEORY the Set of Polynomials Over a Field
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION §1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics. A polynomials is irreducible iff it cannot be factored as a product of polynomials of strictly lower degree. Otherwise, the polynomial is reducible. Every linear polynomial is irreducible, and, when F = C, these are the only ones. When F = R, then the only other irreducibles are quadratics with negative discriminants. However, when F = Q, there are irreducible polynomials of arbitrary degree. As for the integers, we have a division algorithm, which in this case takes the form that, if f(x) and g(x) are two polynomials, then there is a quotient q(x) and a remainder r(x) whose degree is less than that of g(x) for which f(x) = q(x)g(x) + r(x) . The greatest common divisor of two polynomials f(x) and g(x) is a polynomial of maximum degree that divides both f(x) and g(x). It is determined up to multiplication by a constant, and every common divisor divides the greatest common divisor. These correspond to similar results for the integers and can be established in the same way. One can determine a greatest common divisor by the Euclidean algorithm, and by going through the equations in the algorithm backward arrive at the result that there are polynomials u(x) and v(x) for which gcd (f(x), g(x)) = u(x)f(x) + v(x)g(x) . -
Selecting Polynomials for the Function Field Sieve
Selecting polynomials for the Function Field Sieve Razvan Barbulescu Université de Lorraine, CNRS, INRIA, France [email protected] Abstract The Function Field Sieve algorithm is dedicated to computing discrete logarithms in a finite field Fqn , where q is a small prime power. The scope of this article is to select good polynomials for this algorithm by defining and measuring the size property and the so-called root and cancellation properties. In particular we present an algorithm for rapidly testing a large set of polynomials. Our study also explains the behaviour of inseparable polynomials, in particular we give an easy way to see that the algorithm encompass the Coppersmith algorithm as a particular case. 1 Introduction The Function Field Sieve (FFS) algorithm is dedicated to computing discrete logarithms in a finite field Fqn , where q is a small prime power. Introduced by Adleman in [Adl94] and inspired by the Number Field Sieve (NFS), the algorithm collects pairs of polynomials (a; b) 2 Fq[t] such that the norms of a − bx in two function fields are both smooth (the sieving stage), i.e having only irreducible divisors of small degree. It then solves a sparse linear system (the linear algebra stage), whose solutions, called virtual logarithms, allow to compute the discrete algorithm of any element during a final stage (individual logarithm stage). The choice of the defining polynomials f and g for the two function fields can be seen as a preliminary stage of the algorithm. It takes a small amount of time but it can greatly influence the sieving stage by slightly changing the probabilities of smoothness. -
Effective Noether Irreducibility Forms and Applications*
Appears in Journal of Computer and System Sciences, 50/2 pp. 274{295 (1995). Effective Noether Irreducibility Forms and Applications* Erich Kaltofen Department of Computer Science, Rensselaer Polytechnic Institute Troy, New York 12180-3590; Inter-Net: [email protected] Abstract. Using recent absolute irreducibility testing algorithms, we derive new irreducibility forms. These are integer polynomials in variables which are the generic coefficients of a multivariate polynomial of a given degree. A (multivariate) polynomial over a specific field is said to be absolutely irreducible if it is irreducible over the algebraic closure of its coefficient field. A specific polynomial of a certain degree is absolutely irreducible, if and only if all the corresponding irreducibility forms vanish when evaluated at the coefficients of the specific polynomial. Our forms have much smaller degrees and coefficients than the forms derived originally by Emmy Noether. We can also apply our estimates to derive more effective versions of irreducibility theorems by Ostrowski and Deuring, and of the Hilbert irreducibility theorem. We also give an effective estimate on the diameter of the neighborhood of an absolutely irreducible polynomial with respect to the coefficient space in which absolute irreducibility is preserved. Furthermore, we can apply the effective estimates to derive several factorization results in parallel computational complexity theory: we show how to compute arbitrary high precision approximations of the complex factors of a multivariate integral polynomial, and how to count the number of absolutely irreducible factors of a multivariate polynomial with coefficients in a rational function field, both in the complexity class . The factorization results also extend to the case where the coefficient field is a function field. -
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. -
Positivity Conditions for Cubic, Quartic and Quintic Polynomials Arxiv:2008.10922V10 [Math.GM] 18 Sep 2020
Positivity Conditions for Cubic, Quartic and Quintic Polynomials Liqun Qi,∗ Yisheng Song,y and Xinzhen Zhang,z August 17, 2021 Abstract We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary and sufficient condition for a quartic polynomial to be nonnegative for all posi- tive reals. From this, we derive a necessary and sufficient condition for a quartic polynomial to be nonnegative and positive for all reals. Our condition explic- itly exhibits the scope and role of some coefficients, and has strong geometrical meaning. In the interior of the nonnegativity region for all reals, there is an ap- pendix curve. The discriminant is zero at the appendix, and positive in the other part of the interior of the nonnegativity region. By using the Sturm sequences, we present a necessary and sufficient condition for a quintic polynomial to be positive and nonnegative for all positive reals. We show that for polynomials of a fixed even degree higher than or equal to four, if they have no real roots, then their discriminants take the same sign, which depends upon that degree only, except on an appendix set of dimension lower by two, where the discriminants attain zero. arXiv:2008.10922v10 [math.GM] 18 Sep 2020 Key words. Cubic polynomials, quartic polynomials, quintic polynomials, the Sturm theorem, discriminant, appendix. AMS subject classifications. -
Quartic Equation of General Form
EqWorld http://eqworld.ipmnet.ru Exact Solutions > Algebraic Equations and Systems of Algebraic Equations > Algebraic Equations > Quartic Equation of General Form 8. ax4 + bx3 + cx2 + dx + e = 0 (a ≠ 0). Quartic equation of general form. 1±. Reduction to an incomplete equation. The quartic equation in question is reduced to an incom- plete equation y4 + py2 + qy + r = 0.(1) with the change of variable b x = y − 4a 2±. Decartes–Euler solution. The roots of the incomplete equation (1) are given by 1 ¡p p p ¢ 1 ¡p p p ¢ y1 = z1 + z2 + z3 , y2 = z1 − z2 − z3 , 2 2 2 1 ¡ p p p ¢ 1 ¡ p p p ¢ ( ) y3 = 2 − z1 + z2 − z3 , y4 = 2 − z1 − z2 + z3 , where z1, z2, z3 are roots of the cubic equation z3 + 2pz2 + (p2 − 4r)z − q2 = 0,(3) which is called the cubic resolvent of equation (1). The signs of the roots in (2) are chosen so that p p p z1 z2 z3 = −q. The roots of the incomplete quartic equation (1) are determined by the roots of the cubic resolvent (3); see the table below. TABLE Relation between the roots of the incomplete quartic equation and the roots of its cubic resolvent Cubic resolvent (3) Quartic equation (1) All roots are real and positive* Four real roots All roots are real, Two pairs of complex conjugate roots one positive and two negative* One roots is positive Two real and two complex conjugate roots and two roots are complex conjugate 2 * By Vieta’s theorem, the product of the roots z1, z2, z3 is equal to q ≥ 0. -
Generation of Irreducible Polynomials from Trinomials Over GF(2). I
INFORMATION AND CONTROL 30, 396-'407 (1976) Generation of Irreducible Polynomials from Trinomials over GF(2). I B. G. BAJOGA AND rvV. J. WALBESSER Department of Electrical Engineering, Ahmadu Bello University, Zaria, Nigeria Methods of generating irreducible polynomials from a given minimal polynomial are known. However, when dealing with polynomials of large degrees many of these methods are laborious, and computers have to be used. In this paper the problem of generating irreducible polynomials from trinomials is investigated. An efficient technique of computing the minimum polynomial of c~k over GF(2) for certain values of k, when the minimum polynomial of c~ is of the form x m 4- x + 1, is developed, and explicit formulae are given. INTRODUCTION The generation of irreducible polynomials over GF(2) has been a subject of a number of investigations mainly because these polynomials are important not only in the study of linear sequencies but also in BCH coding and decoding. Many results have been obtained (Albert, 1966; Daykin, 1960). Computational methods for generating minimal polynomials from a given irreducible polynomial have been developed. These have been described by Berlekamp (1968), Golomb (1967), and Lempel (1971), among others. Berlekamp observed that all these methods are helpful for hand calculation only if the minimal polynomial from which others are generated is of low degree. He further pointed out that it proves easiest to compute polynomials of large degree by computer using the matrix method. In addition these methods use algorithms. Seldom do they provide results of a general nature. However, utilizing the underlying ideas of the matrix method, some general results on generating irreducible polynomials from a large class of trinomials are derived in this paper. -
Optimal Irreducible Polynomials for GF(2M) Arithmetic
Optimal Irreducible Polynomials for GF(2m) arithmetic Michael Scott School of Computing Dublin City University GF(2m) polynomial representation A polynomial with coefficients either 0 or 1 (m is a small prime) Stored as an array of bits, of length m, packed into computer words Addition (and subtraction) – easy – XOR. No reduction required as bit length does not increase. GF(2m) arithmetic 1 Squaring, easy, simply insert 0 between coefficients. Example 110101 → 10100010001 Multiplication – artificially hard as instruction sets do not support “binary polynomial” multiplication, or “multiplication without carries” – which is actually simpler in hardware than integer multiplication! Really annoying! GF(2m) arithmetic 2 So we use Comb or Karatsuba methods… Squaring or multiplication results in a polynomial with 2m-1 coefficients. This must be reduced with respect to an irreducible polynomial, to yield a field element of m bits. For example for m=17, x17+x5+1 GF(2m) arithmetic 3 This trinomial has no factors (irreducible) Reduction can be performed using shifts and XORs x17+x5+1 = 100000000000100001 Example – reduce 10100101010101101010101 GF(2m) arithmetic 4 10100101010101101010101 100000000000100001 ⊕ 00100101010110000110101 ← 100101010110000110101 100000000000100001 ⊕ 000101010110100111101 ← 101010110100111101 100000000000100001 ⊕ 001010110100011100 ← 1010110100011100 → result! Reduction in software - 1 Consider the standard pentanomial x163+x7+x6+x3+1 Assume value to be reduced is represented as 11 32-bit words g[.] To -
Analytical Formula for the Roots of the General Complex Cubic Polynomial Ibrahim Baydoun
Analytical formula for the roots of the general complex cubic polynomial Ibrahim Baydoun To cite this version: Ibrahim Baydoun. Analytical formula for the roots of the general complex cubic polynomial. 2018. hal-01237234v2 HAL Id: hal-01237234 https://hal.archives-ouvertes.fr/hal-01237234v2 Preprint submitted on 17 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Analytical formula for the roots of the general complex cubic polynomial Ibrahim Baydoun1 1ESPCI ParisTech, PSL Research University, CNRS, Univ Paris Diderot, Sorbonne Paris Cit´e, Institut Langevin, 1 rue Jussieu, F-75005, Paris, France January 17, 2018 Abstract We present a new method to calculate analytically the roots of the general complex polynomial of degree three. This method is based on the approach of appropriated changes of variable involving an arbitrary parameter. The advantage of this method is to calculate the roots of the cubic polynomial as uniform formula using the standard convention of the square and cubic roots. In contrast, the reference methods for this problem, as Cardan-Tartaglia and Lagrange, give the roots of the cubic polynomial as expressions with case distinctions which are incorrect using the standar convention.