DOCSLIB.ORG

Galois Theory Through Exercises

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Galois Theory Through Exercises Contents 1 Solving algebraic equations ............................................ 1 2 Field extensions ....................................................... 9 3 Polynomials and irreducibility ......................................... 13 4 Algebraic extensions ................................................... 17 5 Splitting fields. Finite fields ............................................ 23 6 Automorphism groups of fields. Galois groups ......................... 29 7 Normal extensions ..................................................... 35 8 Separable extensions ................................................... 39 9 Galois extensions ...................................................... 43 10 Cyclotomic extensions ................................................. 51 11 Galois modules ........................................................ 57 12 Solvable groups ........................................................ 63 13 Solvability of equations ................................................ 67 X Contents 14 Geometric constructions ............................................... 71 15 Computing Galois groups .............................................. 75 16 Supplementary problems .............................................. 83 17 Proofs of the theorems ................................................ 93 18 Hints and answers ..................................................... 135 19 Examples and selected solutions ....................................... 163 APPENDIX: Groups, rings and fields ..................................... 221 A.1 Equivalence relations................................................. 221 A.2 Groups............................................................. 222 A.3 Rings.............................................................. 231 A.4 Fields.............................................................. 237 A.5 Chinese Remainder Theorem.......................................... 241 A.6 Polynomial rings.................................................... 241 A.7 Modules over rings.................................................. 243 A.8 Group actions on sets................................................ 246 A.9 Permutations....................................................... 251 A.10 Some arithmetical functions.......................................... 254 A.11 Symmetric polynomials.............................................. 257 A.12 Roots of unity...................................................... 259 A.13 Transitive subgroups of permutation groups............................. 259 A.14 Zorn's Lemma...................................................... 262 A.15 Dual abelian groups................................................. 263 References ................................................................. 264 List of notations ........................................................... 265 Index ...................................................................... 268 Contents XI 1 Solving algebraic equations An algebraic equation of degree n with complex coefficients is an equation: n n−1 f(X) = a0X + a1X + ··· + an−1X + an = 0; where ai 2 C, n ≥ 0 and a0 6= 0 (if n = 0, f(X) is a constant polynomial). According to the Fundamental theorem of algebra (proved by C.F. Gauss1 in 1799), if f(X) is non- constant polynomial, then the equation f(X) = 0 has n complex solutions (roots) x1; : : : ; xn and f(X) = a0(X − x1) ··· (X − xn). Some of the xi may be equal and the number of occurrences of xi as a root of f(X) = 0 is called its multiplicity. One of the main mathematical problems before 18th century was to express solutions of al- gebraic equations through their coefficients using the four arithmetical operations (addition, subtraction, multiplication, division) and radicals. Such possibility was known for n = 1; 2 since antiquity, and for n = 3; 4 since 15th century. Finding similar expressions for equa- tions of degree 5 was an unachievable task. The idea that such formulae may not exist was a motivation for the works of many mathematicians during 18th century and finally lead to the results, which are a part of the theory developed by Evariste´ Galois in the beginning of 19th century. In this chapter, we look at some methods of solving the equations when general formulae exist. We can always assume that a0 = 1 (if not, divide both sides of the equation by a0 6= 0). We recall how to solve quadratic equations and afterwards discuss the cases of cubic and quartic equations. 1 Johann Carl Friedrich Gauss (1777 { 1855) was a German mathematician { one of the most prominent in the history of the subject. He left very important contributions in many parts of mathematics and many other natural sciences including number theory, algebra, statistics, anal- ysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, astronomy and optics. 1.1 Quadratic equations. A quadratic equation f(X) = 0 has 2 roots x1; x2 and f(X) = 2 a0X + a1X + a2 = a0(X − x1)(X − x2). Comparing the coefficients on the left and right in this equality, we get the Vieta formulae2: a1 x1 + x2 = − a0 a2 x1x2 = a0 Denoting a1 = p and a2 = q, we get an equivalent equation: a0 a0 X2 + pX + q = 0 p and solve it by transforming to an equation with roots xi + 2 whose sum is 0, that is, replacing the equation above by the equivalent expression in which X2 + pX is completed to a square: p2 p2 X + + q − = 0: 2 4 Solving, we get the two solutions: r r p p2 p p2 x = − − − q and x = − + − q: 1 2 4 2 2 4 2 2 2 The number ∆ = p − 4q = (x1 + x2) − 4x1x2 = (x1 − x2) is called the discriminant of 2 the polynomial X + pX + q (see p. 258). It is non-zero if and only if the roots x1; x2 are different. 1.2 Cubic equations. A cubic equation f(X) = 0 has 3 roots x1; x2; x3 and f(X) = 3 2 a0X + a1X + a2X + a3 = a0(X − x1)(X − x2)(X − x3). Comparing the coefficients on the left and right in this equality, we get the Vieta formulae: a1 x1 + x2 + x3 = − ; a0 a2 x1x2 + x2x3 + x1x3 = ; (1.1) a0 a3 x1x2x3 = − ; a0 2 Franciscus Vieta, in Franch Fran¸coisVi`ete,(1540{1603) was a Franch mathematician who intro- duced modern notation in connection with algebraic equations. 2 a1 2 The equation having the roots xi + , where i = 1; 2; 3, has the coefficient of x equal to 0, 3a0 since the sum of these 3 numbers is zero. Technically, we get such an equation substituting X = Y − a1 in the given cubic equation: 3a0 3 2 a0X + a1X + a2X + a3 = 3 2 a1 a1 a1 3 a0 Y − + a1 Y − + a2 Y − + a3 = a0(Y + pY + q) 3a0 3a0 3a0 where p and q are easily computable coefficients (it is not necessary to remember these formulae, since it is easier to remember the substitution and to transform each time, when it is necessary). Thus we can start with an equation: X3 + pX + q = 0: (1.2) We compare the last equality with the well-known identity: (a + b)3 − 3ab(a + b) − (a3 + b3) = 0 (1.3) and choose a; b in such a way that: p = −3ab; q = −(a3 + b3): (1.4) If a; b are so chosen, then evidently x = a + b is a solution of the equation (1.2). Since a3 + b3 = −q; p3 a3b3 = − ; 27 a3; b3 are solutions of the quadratic equation: p3 t2 + qt − = 0: 27 Solving this equation gives a3; b3. Then we choose a; b, which satisfy (1.4) and get x = a + b. An (impressing) formula, which hardly needs to be memorized (it is easier to start \from the beginning", that is, from the identity (1.3), than to remember it) is thus the following: 3 s s r 2 3 r 2 3 3 q q p 3 q q p x = a + b = − + + + − − + (1.5) 1 2 4 27 2 4 27 We leave a discussion of different choices of signs of the roots to exercises, but notice that in order to solve a cubic equation, it is sufficient to find one root x1 and solve a quadratic equation after dividing the cubic by x − x1. The above expression (in different notations) was first published by Gerolamo Cardano3 in his book \Ars Magna" in 1545 and is known today as Cardano's formula. However, the history surrounding the formula is very involved and several Italian mathematicians knew the method before Cardano (notably, Scipione del Ferro, Antonio Fiore and Niccol´oTartaglia about 20 years earlier that the appearance of Cardano's book ). 3 2 2 The discriminant of the cubic (1.2) is ∆ = −(4p + 27q ) = [(x1 − x2)(x2 − x3)(x3 − x1)] . See p. 258 and Ex. 1.3 below. 4 3 2 1.3 Quartic equations. Such an equation f(X) = a0X + a1X + a2X + a3X + a4 = a0(x − x1)(X − x2)(X − x3)(X − x4) has 4 roots x1; x2; x3; x4. Similarly as for quadratic and cubic equations, we can write down the Vieta formulae expressing relations between the roots and the coefficients of the equation (we leave it as an exercise). Solving a quartic equation (we assume a0 = 1 and denote the remaining coefficients avoiding indices): X4 + mX3 + pX2 + qX + r = 0; (1.6) m it is not important to assume that m = 0 (but it could be achieved substituting X − 4 instead of X similarly as in the quadratic and cubic cases). We simply complement X4+mX3 in such a way that we get a square of a quadratic polynomial and when it is done, we find a parameter λ so that the quadratic polynomial in square brackets m λ2 m2 mλ λ2 X2 + X + − + λ − p X2 + − q X + − r = 0 (1.7) 2 2 4 2 4 is a square of a first degree polynomial. This is achieved when the discriminant of the quadratic polynomial in the square bracket equals 0 (see Ex. 1.8), that is, mλ 2 m2 λ2 − q − 4 + λ − p − r = 0 (1.8) 2 4 4 This is a cubic equation with respect to λ. Solving it, we get a root λ, which gives a possibility to factorize the quartic polynomial (1.7) into a product of two quadratic polynomials and 3 Girolamo Cardano (1501 { 1576) was an Italian mathematician
Load more

Recommended publications
  • MATH5725 Lecture Notes∗ Typed by Charles Qin October 2007
    MATH5725 Lecture Notes∗ Typed By Charles Qin October 2007 1 Genesis Of Galois Theory Definition 1.1 (Radical Extension). A field extension K/F is radical if there is a tower of ri field extensions F = F0 ⊆ F1 ⊆ F2 ⊆ ... ⊆ Fn = K where Fi+1 = Fi(αi), αi ∈ Fi for some + ri ∈ Z . 2 Splitting Fields Proposition-Definition 2.1 (Field Homomorphism). A map of fields σ : F −→ F 0 is a field homomorphism if it is a ring homomorphism. We also have: (i) σ is injective (ii) σ[x]: F [x] −→ F 0[x] is a ring homomorphism where n n X i X i σ[x]( fix ) = σ(fi)x i=1 i=1 Proposition 2.1. K = F [x]/hp(x)i is a field extension of F via composite ring homomor- phism F,−→ F [x] −→ F [x]/hp(x)i. Also K = F (α) where α = x + hp(x)i is a root of p(x). Proposition 2.2. Let σ : F −→ F 0 be a field isomorphism (a bijective field homomorphism). Let p(x) ∈ F [x] be irreducible. Let α and α0 be roots of p(x) and (σp)(x) respectively (in appropriate field extensions). Then there is a field extensionσ ˜ : F (α) −→ F 0(α0) such that: (i)σ ˜ extends σ, i.e.σ ˜|F = σ (ii)σ ˜(α) = α0 Definition 2.1 (Splitting Field). Let F be a field and f(x) ∈ F [x]. A field extension K/F is a splitting field for f(x) over F if: ∗The following notes were based on Dr Daniel Chan’s MATH5725 lectures in semester 2, 2007 1 (i) f(x) factors into linear polynomials over K (ii) K = F (α1, α2, .
    [Show full text]
  • Solving Solvable Quintics
    mathematics of computation volume 57,number 195 july 1991, pages 387-401 SOLVINGSOLVABLE QUINTICS D. S. DUMMIT Abstract. Let f{x) = x 5 +px 3 +qx 2 +rx + s be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if f(x) is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group F20 of order 20 in the symmetric group S5). When f(x) is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order. 1. Introduction It is well known that an irreducible quintic with coefficients in the rational numbers Q is solvable by radicals if and only if its Galois group is contained in the Frobenius group F20 of order 20, i.e., if and only if the Galois group is isomorphic to F20 , to the dihedral group DXQof order 10, or to the cyclic group Z/5Z. (More generally, for any prime p, it is easy to see that a solvable subgroup of the symmetric group S whose order is divisible by p is contained in the normalizer of a Sylow p-subgroup of S , cf. [1].) The purpose here is to give a criterion for the solvability of such a general quintic in terms of the existence of a rational root of an explicit associated resolvent sextic polynomial, and when this is the case, to give formulas for the roots analogous to Cardano's formulas for the general cubic and quartic polynomials (cf.
    [Show full text]
  • Lecture Notes in Galois Theory
    Lecture Notes in Galois Theory Lectures by Dr Sheng-Chi Liu Throughout these notes, signifies end proof, and N signifies end of example. Table of Contents Table of Contents i Lecture 1 Review of Group Theory 1 1.1 Groups . 1 1.2 Structure of cyclic groups . 2 1.3 Permutation groups . 2 1.4 Finitely generated abelian groups . 3 1.5 Group actions . 3 Lecture 2 Group Actions and Sylow Theorems 5 2.1 p-Groups . 5 2.2 Sylow theorems . 6 Lecture 3 Review of Ring Theory 7 3.1 Rings . 7 3.2 Solutions to algebraic equations . 10 Lecture 4 Field Extensions 12 4.1 Algebraic and transcendental numbers . 12 4.2 Algebraic extensions . 13 Lecture 5 Algebraic Field Extensions 14 5.1 Minimal polynomials . 14 5.2 Composites of fields . 16 5.3 Algebraic closure . 16 Lecture 6 Algebraic Closure 17 6.1 Existence of algebraic closure . 17 Lecture 7 Field Embeddings 19 7.1 Uniqueness of algebraic closure . 19 Lecture 8 Splitting Fields 22 8.1 Lifts are not unique . 22 Notes by Jakob Streipel. Last updated December 6, 2019. i TABLE OF CONTENTS ii Lecture 9 Normal Extensions 23 9.1 Splitting fields and normal extensions . 23 Lecture 10 Separable Extension 26 10.1 Separable degree . 26 Lecture 11 Simple Extensions 26 11.1 Separable extensions . 26 11.2 Simple extensions . 29 Lecture 12 Simple Extensions, continued 30 12.1 Primitive element theorem, continued . 30 Lecture 13 Normal and Separable Closures 30 13.1 Normal closure . 31 13.2 Separable closure . 31 13.3 Finite fields .
    [Show full text]
  • Casus Irreducibilis and Maple
    48 Casus irreducibilis and Maple Rudolf V´yborn´y Abstract We give a proof that there is no formula which uses only addition, multiplication and extraction of real roots on the coefficients of an irreducible cubic equation with three real roots that would provide a solution. 1 Introduction The Cardano formulae for the roots of a cubic equation with real coefficients and three real roots give the solution in a rather complicated form involving complex numbers. Any effort to simplify it is doomed to failure; trying to get rid of complex numbers leads back to the original equation. For this reason, this case of a cubic is called casus irreducibilis: the irreducible case. The usual proof uses the Galois theory [3]. Here we give a fairly simple proof which perhaps is not quite elementary but should be accessible to undergraduates. It is well known that a convenient solution for a cubic with real roots is in terms of trigono- metric functions. In the last section we handle the irreducible case in Maple and obtain the trigonometric solution. 2 Prerequisites By N, Q, R and C we denote the natural numbers, the rationals, the reals and the com- plex numbers, respectively. If F is a field then F [X] denotes the ring of polynomials with coefficients in F . If F ⊂ C is a field, a ∈ C but a∈ / F then there exists a smallest field of complex numbers which contains both F and a, we denote it by F (a). Obviously it is the intersection of all fields which contain F as well as a.
    [Show full text]
  • Arxiv:2010.01281V1 [Math.NT] 3 Oct 2020 Extensions
    CONSTRUCTIONS USING GALOIS THEORY CLAUS FIEKER AND NICOLE SUTHERLAND Abstract. We describe algorithms to compute fixed fields, minimal degree splitting fields and towers of radical extensions using Galois group computations. We also describe the computation of geometric Galois groups and their use in computing absolute factorizations. 1. Introduction This article discusses some computational applications of Galois groups. The main Galois group algorithm we use has been previously discussed in [FK14] for number fields and [Sut15, KS21] for function fields. These papers, respectively, detail an algorithm which has no degree restrictions on input polynomials and adaptations of this algorithm for polynomials over function fields. Some of these details are reused in this paper and we will refer to the appropriate sections of the previous papers when this occurs. Prior to the development of this Galois group algorithm, there were a number of algorithms to compute Galois groups which improved on each other by increasing the degree of polynomials they could handle. The lim- itations on degree came from the use of tabulated information which is no longer necessary. Galois Theory has its beginning in the attempt to solve polynomial equa- tions by radicals. It is reasonable to expect then that we could use the computation of Galois groups for this purpose. As Galois groups are closely connected to splitting fields, it is worthwhile to consider how the computa- tion of a Galois group can aid the computation of a splitting field. In solving a polynomial by radicals we compute a splitting field consisting of a tower of radical extensions. We describe an algorithm to compute splitting fields in general as towers of extensions using the Galois group.
    [Show full text]
  • Graduate Texts in Mathematics 204
    Graduate Texts in Mathematics 204 Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer Science+Business Media, LLC Graduate Texts in Mathematics TAKEUTIlZARING.lntroduction to 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nded. 2 OxrOBY. Measure and Category. 2nd ed. 35 Al.ExANDERIWERMER. Several Complex 3 SCHAEFER. Topological Vector Spaces. Variables and Banach Algebras. 3rd ed. 2nded. 36 l<ELLEyINAMIOKA et al. Linear Topological 4 HILTON/STAMMBACH. A Course in Spaces. Homological Algebra. 2nd ed. 37 MONK. Mathematical Logic. 5 MAc LANE. Categories for the Working 38 GRAUERT/FRlTZSCHE. Several Complex Mathematician. 2nd ed. Variables. 6 HUGHEs/Pn>ER. Projective Planes. 39 ARVESON. An Invitation to C*-Algebras. 7 SERRE. A Course in Arithmetic. 40 KEMENY/SNELLIKNAPP. Denumerable 8 T AKEUTIlZARING. Axiomatic Set Theory. Markov Chains. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 41 APOSTOL. Modular Functions and and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN. A Course in Simple Homotopy 2nded. Theory. 42 SERRE. Linear Representations of Finite II CONWAY. Functions of One Complex Groups. Variable I. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous 12 BEALS. Advanced Mathematical Analysis. Functions. 13 ANDERSoN/FULLER. Rings and Categories 44 KENDIG. Elementary Algebraic Geometry. of Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed. 14 GoLUBITSKy/GUILLEMIN. Stable Mappings 46 LoEVE. Probability Theory II. 4th ed. and Their Singularities. 47 MorSE. Geometric Topology in 15 BERBERIAN. Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SACHs/WU. General Relativity for 16 WINTER. The Structure of Fields. Mathematicians. 17 ROSENBLATT.
    [Show full text]
  • 25 Radical Extensions This Section Contains Two Classic Applications of Galois Theory
    25 Radical extensions This section contains two classic applications of Galois theory. Thefirst application, the problem of the constructibility of points in the plane raised by Greek mathematicians, shows that the underlying questions do not need to refer tofield extensions or auto- morphism groups in any way. The second application, solving polynomial equations by extracting roots, is the problem that, in a way, gave rise to Galois theory. � Construction problems In Greek mathematics,13 people used to constructfigures with a straightedge and com- pass. Here one repeatedly enlarges a given setX of at least two points in the plane by adding to it those points that can be constructed as intersections of lines and circles defined in terms of the given points. The question is, for givenX, whether certain points in the plane can be obtained fromX infinitely many construction steps. More formally, a construction step starting out from a subsetX of the plane consists of replacing the setX with the set (X) of all pointsP obtained by applying F the following algorithm: 1. Pick points a, b, c, d X witha=b andc=d. ∈ � � 2. Let� ab be either the line througha andb or the circle througha with centerb. Likewise, let� cd be either the line throughc andd or the circle throughc with centerd. 3. If� and� do not coincide, pickP � � . ab cd ∈ ab ∩ cd b a c a b d b c c d a d In order to perform step 1 of the algorithm,X needs to contain at least two points. In this case, pickingc=a andd=b shows that we haveX (X).
    [Show full text]
  • Unsolvability of the Quintic Formalized in Dependent Type Theory Sophie Bernard, Cyril Cohen, Assia Mahboubi, Pierre-Yves Strub
    Unsolvability of the Quintic Formalized in Dependent Type Theory Sophie Bernard, Cyril Cohen, Assia Mahboubi, Pierre-Yves Strub To cite this version: Sophie Bernard, Cyril Cohen, Assia Mahboubi, Pierre-Yves Strub. Unsolvability of the Quintic For- malized in Dependent Type Theory. ITP 2021 - 12th International Conference on Interactive Theorem Proving, Jun 2021, Rome / Virtual, France. hal-03136002v4 HAL Id: hal-03136002 https://hal.inria.fr/hal-03136002v4 Submitted on 2 May 2021 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. Unsolvability of the Quintic Formalized in Dependent Type Theory Sophie Bernard Université Côte d’Azur, Inria, France Cyril Cohen  Université Côte d’Azur, Inria, France Assia Mahboubi Inria, France Vrije Universiteit Amsterdam, The Netherlands Pierre-Yves Strub École polytechnique, France Abstract In this paper, we describe an axiom-free Coq formalization that there does not exists a general method for solving by radicals polynomial equations of degree greater than 4. This development includes a proof of Galois’ Theorem of the equivalence between solvable extensions and extensions solvable by radicals. The unsolvability of the general quintic follows from applying this theorem to a well chosen polynomial with unsolvable Galois group.
    [Show full text]
  • Galois Theory and the Insolvability of the Quintic Equation Daniel Franz
    Galois Theory and the Insolvability of the Quintic Equation Daniel Franz 1. Introduction Polynomial equations and their solutions have long fascinated math- ematicians. The solution to the general quadratic polynomial ax2 + bx + c = 0 is the well known quadratic formula: p −b ± b2 − 4ac x = : 2a This solution was known by the ancient Greeks and solutions to gen- eral cubic and quartic equations were discovered in the 16th century. The important property of all these solutions is that they are solu- tions \by radicals"; that is, x can be calculated by performing only elementary operations and taking roots. After solutions by radicals were discovered for cubic and quartic equations, it was assumed that such solutions could be found polynomials of degree n for any natu- ral number n. The next obvious step then was to find a solution by radicals to the general fifth degree polynomial, the quintic equation. The answer to this problem continued to elude mathematicians until, in 1824, Niels Abel showed that there existed quintics which did not have a solution by radicals. During the time Abel was working in Nor- way on this problem, in the early 18th century, Evariste Galois was also investigating solutions to polynomials in France. Galois, who died at the age of 20 in 1832 in a duel, studied the permutations of the roots of a polynomial, and used the structure of this group of permuta- tions to determine when the roots could be solved in terms of radicals; such groups are called solvable groups. In the course of his work, he became the first mathematician to use the word \group" and defined a normal subgroup.
    [Show full text]
  • A Short History of Complex Numbers
    A Short History of Complex Numbers Orlando Merino University of Rhode Island January, 2006 Abstract This is a compilation of historical information from various sources, about the number i = √ 1. The information has been put together for students of Complex Analysis who − are curious about the origins of the subject, since most books on Complex Variables have no historical information (one exception is Visual Complex Analysis, by T. Needham). A fact that is surprising to many (at least to me!) is that complex numbers arose from the need to solve cubic equations, and not (as it is commonly believed) quadratic equations. These notes track the development of complex numbers in history, and give evidence that supports the above statement. 1. Al-Khwarizmi (780-850) in his Algebra has solution to quadratic equations of various types. Solutions agree with is learned today at school, restricted to positive solutions [9] Proofs are geometric based. Sources seem to be greek and hindu mathematics. According to G. J. Toomer, quoted by Van der Waerden, Under the caliph al-Ma’mun (reigned 813-833) al-Khwarizmi became a member of the “House of Wisdom” (Dar al-Hikma), a kind of academy of scientists set up at Baghdad, probably by Caliph Harun al-Rashid, but owing its preeminence to the interest of al-Ma’mun, a great patron of learning and scientific investigation. It was for al-Ma’mun that Al-Khwarizmi composed his astronomical treatise, and his Algebra also is dedicated to that ruler 2. The methods of algebra known to the arabs were introduced in Italy by the Latin transla- tion of the algebra of al-Khwarizmi by Gerard of Cremona (1114-1187), and by the work of Leonardo da Pisa (Fibonacci)(1170-1250).
    [Show full text]
  • Radical Extensions and Galois Groups
    Radical extensions and Galois groups een wetenschappelijke proeve op het gebied van de Natuurwetenschappen, Wiskunde en Informatica Proefschrift ter verkrijging van de graad van doctor aan de Radboud Universiteit Nijmegen op gezag van de Rector Magni¯cus prof. dr. C.W.P.M. Blom, volgens besluit van het College van Decanen in het openbaar te verdedigen op dinsdag 17 mei 2005 des namiddags om 3.30 uur precies door Mascha Honsbeek geboren op 25 oktober 1971 te Haarlem Promotor: prof. dr. F.J. Keune Copromotores: dr. W. Bosma dr. B. de Smit (Universiteit Leiden) Manuscriptcommissie: prof. dr. F. Beukers (Universiteit Utrecht) prof. dr. A.M. Cohen (Technische Universiteit Eindhoven) prof. dr. H.W. Lenstra (Universiteit Leiden) Dankwoord Zoals iedereen aan mijn cv kan zien, is het schrijven van dit proefschrift niet helemaal volgens schema gegaan. De eerste drie jaar wilde het niet echt lukken om resultaten te krijgen. Toch heb ik in die tijd heel ¯jn samengewerkt met Frans Keune en Wieb Bosma. Al klopte ik drie keer per dag aan de deur, steeds maakten ze de tijd voor mij. Frans en Wieb, dank je voor jullie vriendschap en begeleiding. Op het moment dat ik dacht dat ik het schrijven van een proefschrift misschien maar op moest geven, kwam ik in contact met Bart de Smit. Hij keek net een beetje anders tegen de wiskunde waarmee ik bezig was aan en hielp me daarmee weer op weg. Samen met Hendrik Lenstra kwam hij met het voorstel voor het onderzoek dat ik heb uitgewerkt in het eerste hoofdstuk, de kern van dit proefschrift.
    [Show full text]
  • ON the CASUS IRREDUCIBILIS of SOLVING the CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 Jvillanu@Fmu
    ON THE CASUS IRREDUCIBILIS OF SOLVING THE CUBIC EQUATION Jay Villanueva Florida Memorial University Miami, FL 33055 [email protected] I. Introduction II. Cardan’s formulas A. 1 real, 2 complex roots B. Multiple roots C. 3 real roots – the casus irreducibilis III. Examples IV. Significance V. Conclusion ******* I. Introduction We often need to solve equations as teachers and researchers in mathematics. The linear and quadratic equations are easy. There are formulas for the cubic and quartic equations, though less familiar. There are no general methods to solve the quintic and other higher order equations. When we deal with the cubic equation one surprising result is that often we have to express the roots of the equation in terms of complex numbers although the roots are real. For example, the equation – 4 = 0 has all roots real, yet when we use the formula we get . This root is really 4, for, as Bombelli noted in 1550, and , and therefore This is one example of the casus irreducibilis on solving the cubic equation with three real roots. 205 II. Cardan’s formulas The quadratic equation with real coefficients, has the solutions (1) . The discriminant < 0, two complex roots (2) ∆ real double root > 0, two real roots. The (monic) cubic equation can be reduced by the transformation to the form where (3) Using the abbreviations (4) and , we get Cardans’ formulas (1545): (5) The complete solutions of the cubic are: (6) The roots are characterized by the discriminant (7) < 0, one real, two complex roots = 0, multiple roots > 0, three real roots.
    [Show full text]