Galois Groups in the work of Mira Fernandes
GGalois Groups in the work of Mira Fernandes
Amaro Rica da Silva
Centro Multidisciplinar de Astrof´ısica - CENTRA,Departamento de F´ısica, Instituto Superior T´ecnico - IST,Universidade T´ecnica de Lisboa - UTL,Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal,
Aureliano Mira Fernandes was a student at the University of Coimbra from1904 until 1910 when he finished his Mathematics degree. He studied Calculuswith Sid´onio Pais and Analysis with Jos´e Bruno de Cabedo. In March, 1911,he completed his Ph.D. thesis entitled ”Teorias de Galois I-Elementos da teoriados grupos de substitui¸c˜oes” (i) under the orientation of Prof. Souto Rodriguesof the University of Coimbra. He was then invited to be Full Professor at theIST in November the same year 1911.His Ph.D. thesis was presented at a time when group theory ideas were emergingat the forefront of scientific research in many areas in most European countriesbut were not widely known. For instance the first english expositions and trans-lations of Galois theory appear around 1891-1900 but are essentially geared to-wards construction methods of Galois groups (O. Bolza, J. Pierpoint, H. Voigt)and up to 1908 no course on Galois theory was taught at Cambridge or Oxford.As was kindly noted to me by Prof. Paulo Almeida, there were a few Portuguesemathematicians in the late 1900 that studied algebraic equations, such as Prof.Luiz Woodhouse at the Academia Polit´ecnica do Porto, who included the sub-ject in his course Higher Algebra and Analytical Geometry, and the Jornal deSciencias Mathematicas e Astronomicas da Universidade de Coimbra publishedworks on algebraic equations by Martins da Silva (1882) and Whoodhouse (in1885) where reference to results by Galois can be found, but these works do notaddress Galois methods or theory.In his 1911 Ph.D. thesis Mira Fernandes focuses on the results around whichthe group structure behind the solvability theorems of Galois theory reside, anddeals mainly with finite group theory definitions and results with permutationgroup realizations. Although he refers to algebraic equation root finding andGalois theory he does not deal with these methods in his thesis, which he leavesuntil a later publication in 1931, in part II of a work entitled ”Grupos de Sub-stitui¸c˜oes e Resolubilidade Alg´ebrica” published by the Instituto Superior deCom´ercio de Lisboa. The latter publication is definitely of a pedagogical natureby the time it is published, as Mira Fernandes’ interests by then had evolved tothe applications of Lie Groups to general relativistic theories. (i) ”Galois Theories I - Elements of the theory of finite substitution groups” reedited as”Substitution Groups and Algebraic Solvability I” (1929)[Fer29] a r X i v : . [ m a t h . HO ] J a n he following table of contents from these works illustrates the subject matterdealt with in each publication. Galois Theories I - Elements of the theory of finite substitution groups(1911)[Fer10]
Introduction: Algebraic SolvabilityI- Finite Groups: Transitivity and primitivityII- Isomorphism and Group composition: Jordan-H¨older and Sylow theorems, Solvablegroups.III- Abelian GroupsIV- Metacyclic group, General Linear group and the Modular group.V- The structure of the total group and the Alternating group - Possible orders of thesimple groups.VI- Generalization of the concept of isomorphism: linear substitution groups of finite or-der. (ii)
VII- Geometrical representation of finite groups of linear substitutions: groups of regularpolyhedra.
Substitution Groups and Algebraic Solvability II (1931)[Fer31]
I- Algebraic Field-Irreducibility of polynomialsII- Galois Resolvent. Galois Group and propertiesIII- General Resolvent. Structure of the Galois group.IV- Abelian Equations.V- Solvability via radicals.
By the end of the XIX century, and definitely by 1931 the focus of GaloisTheories had shifted towards Number Theory applications by the explorationof extension fields mostly by the German mathematicians following the workof Dedekind. Nowadays one can see that this path to Number Theory can besummarized with the following diagram (cid:18)
Solution of Algebraic EquationsGeometric Construction Problems (cid:19) = ⇒ (cid:18) Theory ofPolynomials (cid:19) = ⇒ (cid:18) CommutativeField Theory (cid:19)
In this presentation we intend to show how the group theory concepts discoveredby Galois are connected to previous work that took centuries to evolve, and howfast they changed the landscape of theoretical and applied mathematics andphysics since its publication. It was the concept of radicals and their use in thesolution of algebraic problems that ultimately led to the concept of fields andGalois group theory. Afterwards a revolution took place as his group theoryconcepts took hold with applications in great many areas in mathematics andphysics.The road that leads to Galois theory is made of contributions by mathematiciansthat were trying to find ways to express the general roots of algebraic equationsusing rational expressions and radicals involving the constant coefficients ofthese equations, as was made by del Ferro, Cardano and Ferrari in the XVI th century for equations of order up to 4. [1] [2] [3] (ii) meaning z → z (cid:48) = p z + qp (cid:48) z + q (cid:48) with p q (cid:48) − p (cid:48) q (cid:54) = 0 th century, and it became apparent that the general equations offifth order or greater were problematic in this respect. First came the realizationthat methods that were used to simplify equations of degree less than four wouldnot work for these higher order equations. [4] Then in the XVIII th century Lagrange recognizes that the key to the solvabilityof some equations is the invariance under permutations of their arguments ofcertain symmetric rational functions of the roots, that he calls resolvents. In1771 permutations were first employed by J.L. Lagrange in his ”R´eflexions sur lar´esolution alg´ebrique des ´equations”. By the same time Vandermonde also usessymmetric rational functions of the roots of algebraic equations (somewhat akinto Lagrange’s resolvents) to find solutions by radicals of cyclotomic equations x p − p = 11 (and of a particular case of an equation ofdegree 5 related to them). [5] [6] At the dawn of the XIX th century Paolo Ruffini in his “Teoria Generale delleEquazioni” attempts a 516 page demonstration that general equations of degreeat least 5 are not solvable by radicals. In 1810, in view of a poor reception bythe mathematical community, Ruffini submits an improved version to the Frenchacademy, but the referees Lagrange, Lacroix and Legendre were so delayed withan answer that Ruffini wrote to the president of the academy to withdraw thesubmitted work.In 1801 Gauss studies cyclotomic equations and achieves important results re-garding the solvability of these equations that would serve as foundation for thetheory of algebraic equations. In the process he solves a 2 000 year-old problemon the construction of a polygon with straight edge and compass. [7] Then in 1824 Niels-Henrik Abel (1802-1829) shows that the general algebraicequation of the fifth degree has no solution via radicals. Soon after he extendsthis result to the nonexistence of a general solution with radicals for algebraicequations of degree greater than four.When he dies in 1829 he was addressing the problem of recognizing if a par-ticular algebraic equation of high degree can be solvable by radicals but hecouldn’t complete his work. Still he was able to show that equations whoseroots x k are rational functions x k = F k ( x ) of a single root x all of whichverify F k ( F j ( x )) = F j ( F k ( x )) are solvable. These equations are now called“Abelian Equations”. It is Abel that introduces the concepts of Field and of
Irreducible Polynomial (iii) over a field. From Gauss, Ruffini and Abel’s workwe conclude that cyclotomic equations of any degree are solvable by radicals.Then comes Galois, who submitted in 1829, at the young age of 18, two papers onthe solution of algebraic equations to the Academy of Sciences, which were lostby Cauchy. In 1830 he again presented a paper on his research to the Academyof Sciences. This was sent to Fourier, who unfortunately died soon after andthat paper was also lost. The 1831 article submitted to Poisson and entitled”Sur les conditions de r´esolubilit´e des ´equations par radicaux” was returned byPoisson as unintelligible. On the eve of his deadly duel in 1832 he writes to his (iii)
A polynomial over a field F is said to be Reducible if it can be expressed as a product oftwo polynomials of lesser degree over the same field. Otherwise it will be called
Irreducible . Galois group of an equation , and this group must have a structure of what isnow known as a solvable group . But in doing so he achieved much more, sincehis method can be extended to fields other that the rationals and the structurethat emerges from the groups can be used to classify field extensions and theirsubfields.Galois discovered a method of finding the group of a given equation, the suc-cessive partial resolvent equations and their associated groups that result fromextending the field of coefficients with the roots of these resolvents. Thesegroups turn out to be subgroups of the original group. Galois shows that whenthe group of an equation with respect to a given field is the identity, then theroots of the equation are members of that field.Application of Galois’ theory to the solution of polynomial equations by rationaloperations and radicals then follows. When the partial resolvent that serves toreduce to a subgroup G the group G of an equation is of the form of a binomialequation x p = a with p prime, then G is a normal subgroup of index p of G . Conversely a normal subgroup G of prime index p of G yields a binomialresolvent x p = a .The basic idea is to show that for each algebraic equation of degree n there is aGalois resolvent V = V ( x , . . . , x n ) which is a rational expression of its roots x i with the property that we can in principle find rational expressions for the x i as x i = f i ( V ) (1)The Galois Resolvent is found to be one of the roots V k of a polynomial equation R ( V ) = 0, called the minimal polynomial , such that substitution of any ofits roots in (1) yields a permutation of the roots x i k = f i ( V k )These permutations form the Galois Group of the equation.A basic example illustrates the method. Consider the quadratic equation in xx + a x + b = 0with two roots x and x . The Galois resolvent in this case is simply V = x − x Taking into consideration the general relations between the roots (cid:88) i =1 x i = x + x = − a ; (cid:88) i
4, the Galois groups S n are never solvable, since their maximal normal subgroup is the alternatinggroup A n of order n !2 , and this type of group for n > n !2 , and n !2 is never prime for n ≥ Finite AbstractGroup as any finite set with an associative composition and a neutral element.His work is largely ignored by the community at the time as matrices andquaternions were new and not well known.By 1866 Serret lectures at the Sorbonne about Galois’s work after Liouvillepublication in the Journal de Math´ematiques. It is said that Serret’s 3 rd editionof the Cours d’Alg´ebre Sup´erieure was so popular that its adoption in France’smathematics curricula for the next 50 years hindered there the divulgation ofthe latest developments in group theory. Serret first studies representations ofsubstitutions by transformations of the form z → z (cid:48) = p z + qp (cid:48) z + q (cid:48) with p q (cid:48) − p (cid:48) q (cid:54) = 0Then in 1868 Jordan starts the first investigations of Infinite Groups with paper“M´emoire sur les groupes de mouvements” after Bravais in 1849 had studiedgroups of motions to determine the possible structure of crystals.In 1870 Jordan publishes the ”Trait´e des substitutions et des ´equations alg´ebriques”where he organizes his work on finite substitution groups and their connectionwith Galois theory. He is the first mathematician to focus on the group the-ory aspects of the work rather than on finding the roots of equations. Jordan6olves Abel’s problem of determining which equations of a given degree are solv-able by radicals. He concludes that the groups of such solvable equations arecommutative, which he then names ”Abelian”.In 1873 Jordan introduces the notion of quotient group . He establishes thaton different composition series of the same group there are always the samenumber of elements and the order of the quotient groups is the same up toordering. In 1889 H¨older shows that the quotient groups in a composition seriesare isomorphic up to ordering (Jordan-H¨older Theorem).It is Jordan who initiates the study of geometric transformations with groupsas he studies infinite groups of translations and rotations and represents substi-tutions by linear transformations of the form x i → x (cid:48) i = n (cid:88) j =1 A ij x j In the late XIX th century, German mathematics was very active pursuing prob-lems in Number Theory. Richard Dedekind (1831 − − − − Split-ting Field of the equation as the smallest field containing the roots and thecoefficients.In 1931 Bartel van der Waerden publishes ”Moderne Algebra” using the lecturesby E. Artin and E. Noether. In 1963 the Feit-Thompson theorem is proven show-ing that a finite group of odd order is necessarily solvable. A very interestingaccount of the development of Galois theory up to Artin’s work can be foundin [Kie71].In the meantime, since the mid-1800 the group concept irrespective of its Galoisconnotations found its way to applications in Differential Geometry throughthe works of Sophus Lie, Felix Klein and many others. Klein’s Erlangen Pro-gram is the best statement of the universality of the group notion, and its use7ia Representation Theory in Physics is the trademark of the XX th centuryphysics, were we associate physical symmetry groups and their irreducible rep-resentations with measurable characteristics of natural systems. Using eitherdiscrete or continuous, differentiable groups and its infinitesimal counterpart,the Lie Algebra, one can understand (or model) such diverse aspects as Spon-taneous Symmetry Breaking and Renormalization Methods [DE84], the reasonfor the structure of the Periodic Table [Ste95], the Selection Rules in Quan-tum Mechanics [DE84] [Ste95] [Ham89], the Relativistic and Non-relativisticDynamical Symmetries [BR86], the Symmetries of Differential Equations andConservation Laws [SW93] [Ste90] [Olv00], the reason why Parity is not a nat-ural symmetry [Ste95] or why all Relativistic Wave Equations are an expressionof projection operators of Induced Representations of the Lorentz or Poincar´egroups [DE84][BR86].Coming back to Mira Fernandes, we can see how close he was in his 1910 thesis tothe formal developments that the finite abstract group theory had brought in thelast decade of the XIX th century. The example groups that he studies (Abelian,Metacyclic, General linear, Modular and Symmetric) are not connected in thiswork to any applications to the resolution of algebraic equations. The resultsthat he exposes are of general nature in finite group theory, but the name“Galois Theories-I” in his thesis indicate already his intention of applying thesegroup theoretical concepts to the theory of algebraic equations, a work that hefinishes with his 1931 initiation text on algebraic solvability of equations in thebook entitled “Grupos de Substitui¸c˜oes e Resolubilidade Alg´ebrica” (iv) [Fer31].A couple of years before, in 1929, he had reedited most of his thesis material aspart I of this book [Fer29].Mira Fernandes’ approach to Galois groups and algebraic equations in his 1931work [Fer31] is a little different in that he introduces the fundamental function φ = n (cid:88) i =1 α i − x i with α an arbitrary constant not a root of the discriminant of his “Galois Re-solvent” R ( y ) = n ! (cid:89) i =1 ( y − φ i ) = 0 (3)where the φ i are n ! expressions obtained from permutations of the roots x i inthe fundamental function φ . He then defines the Galois Group of the algebraicequation f ( x ) = 0 as the set of permutations of the roots x i that transform φ in each of the distinct roots φ i of an irreducible component of (3). He startsby showing that such a resolvent always exists, and proceeds to the theoreticalconstruction of the resolvent polynomial and to prove the theorems in grouptheory that are now associated with the interpretation of solvability by radicalsof algebraic equations. He treats explicitly the general cases for Polynomialequations, and gives a few examples of Abelian equations.It is reasonable to assume that by then his interest in this Galois theory is merelyacademic, and probably motivated by his interest in the life of ´Evariste Galois, (iv) Substitution Groups and Algebraic Solvability th century mathematical physics. Notes [1]Algebraic equations until the XVI th century We can trace back the (numerical) methods for the solution of quadratic and bi-quadraticequations almost 4 000 years as it clear that Babylonians already knew how perform square-root operations since ca. 1900 BD. The following cuneiform tablet shows how to compute √ √ solving what is basically a Pythagorean problem numerically. Babylonian mathemat-ics is a set of numerical recipes for solving day-to-day problems, and even though there aremany examples of training exercises for apprentices, there was never an attempt to formalizethe theory. = + + = 0 . √ . + + = 1 . √ . th century AD (Th´eon of Alexandria). reek dedication to geometrical methods is probably a consequence of the fact that, in the”Elements”, Euclid restricts himself to using straight edge and compass methods only, assuggested by Plato.Greek mathematicians also used geometrical methods to represent incommensurate ratios,which they did not consider numbers, and knew already that an irreducible algebraic equa-tion of third degree over the field of rationals, such as the duplication of the cube ( x = 2)or the trisection of an angle ( x − x − b = 0), cannot be solved with straight edge andcompass only. In fact we know today that few algebraic equations possess roots that may befound in this way.Leonardo de Pisa is responsible for the introduction of Arabian mathematical methods in thewestern world in the XIII th century, and the solution of quadratic equations is then perfectedthrough formulas using radicals. [2]Scipione del Ferro (Bologna, 1462-1526) In 1512 del Ferro solves the general cubic equation.A general cubic equation u + a u + a u + a = 0 can always be reduced to a form x + a x = b by the substitution u = x − a with: a = a − a b = − a + a a − a x = y − z on x + a x = b yields y − z + ( y − z ) ( a − y z ) = b Under the conditions for( y − z )( a − y z ) = 0 ⇐⇒ y z = a y = z del Ferro solves the two-variable system for y and z y − z = by z = (cid:0) a (cid:1) thus obtaining y = b (cid:114)(cid:0) a (cid:1) + (cid:16) b (cid:17) ; z = − b (cid:114)(cid:0) a (cid:1) + (cid:16) b (cid:17) del Ferro never publicizes his findings, sharing it only with one of his students, which latterwill divulge it in the form of a sonnet. [3]Jeronimo Cardano (Pavia,1501-1543) In 1545 Cardano publishes in his ”Ars Magna” the del Ferro-Cardanoformula for the cubic x + a x = b.x = b (cid:115) a
27 + b / − − b (cid:115) a
27 + b / There was at the time great resistance to this formula, not only becauseit used negative numbers, but also because real roots had to be ob-tained by summing what we now know as complex numbers. Cardanonever uses the complex solutions that are implicit in the radicals of y and z when considering the 3-roots of unity ζ k (3) = e π i k = ( − k .In fact y = b (cid:115) a
27 + b ⇒ y = (cid:18) b + (cid:113) a + b (cid:19) / y = − ( − / (cid:18) b + (cid:113) a + b (cid:19) / y = ( − / (cid:18) b + (cid:113) a + b (cid:19) / = − b (cid:115) a
27 + b ⇒ z = (cid:18) − b + (cid:113) a + b (cid:19) / z = − ( − / (cid:18) − b + (cid:113) a + b (cid:19) / z = ( − / (cid:18) − b + (cid:113) a + b (cid:19) / Only in 1732 will Euler show that of the 9 possible combinations of x ij = y i − z j only thosewith y i z j = a are good roots and that the correct formulas are: x k = ζ k (3) (cid:18) b + (cid:113) a + b (cid:19) / − ζ k (3)2 (cid:18) − b + (cid:113) a + b (cid:19) / if a ≥ x k = ζ k (3) (cid:18) b + (cid:113) a + b (cid:19) / + ζ k (3)2 (cid:18) b − (cid:113) a + b (cid:19) / if a < R with roots of the unity and other radicals.Cardano gets his medical doctor degree in 1526 by the University of Padua. In 1534 he startslecturing Mathematics in Milan, but maintains his studies in medicine, astrology and magic.In 1570 he is arrested by the Inquisition on charges of having drawn the horoscope of JesusChrist. He was released but barred from giving any more lectures. [4] E. W. von Tschirnhaus (1651-1708) By the end of the XVII th century Leibnitz and Tschirnhaus wereamong the few that still studied solutions of algebraic equations withradicals. In 1683 Tschirnhaus proposes a method to convert a generalpolynomial equation of degree n P n ( x ) = x n + a x n − + . . . + a n − x + a n = 0into a lower degree polynomial by eliminating x with an auxiliary equa-tion of degree n − xy = x n − + b x n − + . . . + b n − x + b n − (4)Then by canceling the ( n − coefficients of x n ( n − , . . . , x n in the linear combination Q n ( y ) = y n + A y n − + . . . + A n − y + A n = (cid:16) α n ( n − x n ( n − + · · · + α x + α (cid:17) P n ( x )one can solve for the α i and replace these in the remaining n equations to obtain the A k = A k ( a , b ), where a = { a , . . . , a n } and b = { b , . . . , b n − } .Imposing the constraint equations A k ( a , b ) = 0 ( k = 1 , , . . . , n −
1) (5)one could (in principle) obtain the unknown coefficients b = { b k ( a ) } so that A n ( a , b ) = − c n ( a )and Q n ( y ) is the binomial equation Q n ( y ) = y n − c n ( a ) = 0 (6)With the new coefficients b k ( a ) and the roots y n,k ( a ) = e π i kn n (cid:112) c n ( a ) of this binomialequation (6) one obtains from (4) a set of lower order polynomial equations P n − ( x, k ) = x n − + b ( a ) x n − + . . . + b n − ( a ) x + b n − ( a ) − y n,k ( a ) = 0whose roots could be determined by the same method. The goal then is to reach a pointwhere we determine a set of monomials P ( x, k, . . . , m ) = x + β ( a ) − y ,m ( a ) rom which the roots of the original equation can be selected. Unfortunately this method onlyworks for n ≤
4, since for n = 5 the constraint equations (5) are of degree 24 in the b k , thusbeing harder to solve than the original equation, and this worsens with increasing n . Themethod also produces false roots among the genuine ones. [5]Joseph-Louis de Lagrange (1736-1813) Lagrange recognizes that what distinguishes algebraic equations of de-gree n ≤ β ( x , . . . , x n ) that under all the permuta-tions of the n roots yield a small number m < n of distinct expressions β k ( x , . . . , x n ) = β (cid:0) x k , . . . , x k n (cid:1) .There is then a Resolvent Equation R n ( y ) = y m + b y m − + . . . + b m − y + b m = 0 , with coefficients b k = b k ( a ) whose roots Y k ( b ) = β k ( x , . . . , x n ) areprecisely these invariant functions.The Lagrange resolvents are of the form ρ k ( x , . . . , x n ) = n (cid:88) i =1 (cid:16) ζ k ( n ) (cid:17) i x i His invariant functions are β k ( x , . . . , x n ) = (cid:18) n ρ k ( x , . . . , x n ) (cid:19) n For instance, in the cubic equation P ( x ) = x + ax = b , given that the 3-roots of unity obey (cid:16) ζ (3)1 (cid:17) = ζ (3)2 ; (cid:16) ζ (3)2 (cid:17) = ζ (3)1 ; (cid:16) ζ (3)1 (cid:17) = (cid:16) ζ (3)2 (cid:17) = ζ (3)3 = 1we obtain six different ρ k but only two distinct β k since the last one is identically zero. k ρ k ( x i , x i , . . . , x i n ) ρ k ( x i , x i , . . . , x i n ) , ζ x + ζ x + x ζ x + ζ x + x ζ x + ζ x + x ζ x + ζ x + x ζ x + ζ x + x ζ x + ζ x + x (cid:89) i =1 x i +3 ζ (cid:0) x x + x x + x x (cid:1) ++3 ζ (cid:0) x x + x x + x x (cid:1) + (cid:88) i =1 x i (cid:89) i =1 x i +3 ζ (cid:0) x x + x x + x x (cid:1) ++3 ζ (cid:0) x x + x x + x x (cid:1) + (cid:88) i =1 x i x + x + x ( x + x + x ) The Resolvent equation for the cubic is then of degree 2. R n ( y ) = ( y − β ( x i )) ( y − β ( x i )) = y − by − a Its solutions Y ( a, b ) = b + (cid:114)(cid:16) b (cid:17) + (cid:0) a (cid:1) ; Y ( a, b ) = b − (cid:114)(cid:16) b (cid:17) + (cid:0) a (cid:1) then imply ρ = ζ x + ζ x + x = 3 Y ( a, b ) / ρ = ζ x + ζ x + x = 3 Y ( a, b ) / ρ = x + x + x = 0with solutions x = ζ Y ( a, b ) / + ζ Y ( a, b ) / x = ζ Y ( a, b ) / + ζ Y ( a, b ) / x = Y ( a, b ) / + Y ( a, b ) / or n = 5 the smallest number of nontrivial β is 24.Lagrange shows first that for any rational expression F ( x , . . . , x n ) the number of its distinctvalues under all permutations is m = n ! |I F | , where |I F | is the number of permutations thatleave F invariant. Nowadays Lagrange’s theorem states that the order of any subgroup H ofa group G divides the order of G . [6]Al´exandre-Th´eophile Vandermonde (1735-1796) Vandermonde explicitly states that an algebraic expression for the roots of polynomials mustbe ambiguous given that the enumeration of roots is arbitrary.For a polynomial P n ( x ) = x n + a x n − + . . . + a n − x + a n an expression F ( x , . . . , x n ) involving radicals n √ exists such that, depending on the choiceof the radical, each root x k is determined. Furthermore this expression F ( x , . . . , x n ) can bewritten solely in terms of the coefficients a = { a , a , . . . , a n } given that a = − (cid:88) i x i , a = (cid:88) i 14 ( x − x ) (cid:19) / = 12 (cid:18) − a + (cid:16) ζ (2)1 (cid:17) j (cid:113) a − a (cid:19) In view of the difficulties of his method for n > x p − p non-prime its roots may be expressed by radicals if the roots of its prime factorsare, all that remains is the study of cyclotomic equations with p prime. Since ζ = 1 is alwaysa root, one must study then the roots of x p − x − x p − + x p − + . . . + x + 1 = 0 (8)In his method Vandermonde proposes that one finds for a cyclotomic equation x p − m < p symmetric rational functions β i ( ζ , . . . , ζ p ) of the roots of the originalequation such that the β i can be obtained as roots of algebraic equations of degree m .For p prime, the division of expression (8) by x q , with q = p − , and a variable change to Y = x + x expresses equivalent equations of lower degrees. x + x = 0 1 + Y = 01 + x + x + x + x = 0 − Y + Y = 01 + x + x + x + x + x + x = 0 − − Y + Y + Y = 01 + x + x + x + x + x + x + x + x = 0 1 − Y − Y + Y + Y = 01 + x + x + x + x + x + x + x + x + x + x = 0 1 + 3 Y − Y − Y + Y + Y = 0 Vandermonde is the first to find solutions by radicals of the cyclotomic equation x p − p = 11, and as a consequence obtains such solutions for a reduced equation of degree p − = 5. In 1801 Gauss studies cyclotomic equations (i.e. equations for thedivision of the circle) x p − x p − Z m ( x ) = 0 whose degrees m are prime factors of p − 1, andwhose coefficients are rational in the roots of the preceding equationsin the sequence. x p − (cid:89) m | p Z m ( x ) Z m ( x ) = ϕ ( m ) (cid:89) i (cid:16) x − ζ ( m ) i (cid:17) = (cid:89) k | m (cid:16) − x mk (cid:17) µ ( k ) where ϕ ( m ) is the Euler function (counting positive co-primes i of m ) and µ ( k ) is the M¨obiusfunction µ ( k ) = k = 10 k has repeated prime factors.( − n k has all n prime factors distinct.This result shows that some equations of high degree n can be solved by radicals if n is afactor of p − p − p sides can be constructed with straight edge and compass. This is because each of theequations Z i ( x ) = 0 is of degree 2 and each of its roots is so determined. Thus Gauss showedthat all polygons of order p = 2 n + 1 are constructible, and for the first time in 2 000 yearsgives the ruler-and-compass construction of a 17-side polygon. Furthermore he shows thatthe next higher polygon so constructible would have 257 sides, then 65 537, etc., as in thesequence 2 n + 1 = 3 , , , , 65 537 , , . . . [8] ´Evariste Galois (1811-1832) In the following example we illustrate from a known solvable equationthe concepts developed by Galois. Of course the theory works withoutknowing first the solutions of the equations, and it makes use of the-orems relating symmetric functions of the roots, the construction ofthe Galois partial resolvents and its use in determining a compositionseries of subgroups. We can also see how the concept of field exten-sion enters the theory. Notice that even though the theory shows theexistence of Galois Resolvents and how it relates to the constructionof the Galois Group of the equation, it is most of the times extremelyhard to find these functions explicitly, therefore an indirect approachhas to be taken. Galois himself knew that his method was not a way toconstruct explicit representations of the roots of polynomial equationsvia rational expressions of the coefficients in a particular extended field(what nowadays would be called the splitting field).Consider the quartic (biquadratic) equation P ( x ) = x + ax + b = 0 (9)In this case we already know the roots of this equation since we can solve it for y = x as aquadratic equation and then use x i ± = ± √ y i . One possible labeling of these solutions wouldbe x = − x = √ y ; x = − x = √ y but there are 23 other choices. We will find out the Galois group of equation (9) from certainsymmetries associated with the arbitrary choice of indexes for the roots.Set Q ( a, b ) the field of rational expressions in a, b with coefficients in the field Q of rationalnumbers. The field Q ( a, b ) is called an extension of the rational field Q . Then the followingrelations hold in Q ( a, b ) x + x = 0 ; x + x = 0 (10) ow the root set of equation (9) is invariant under the 4! = 24 permutations of their labeling,but only 8 of these permutations will leave the relations (10) invariant, and they are in theset G : G = σ = (1)(2)(3)(4) , σ = (12)(3)(4) , σ = (1)(2)(34) , σ = (12)(34) σ = (13)(24) , σ = (1423) , σ = (1324) , σ = (14)(23) This set is a group under the composition of permutations σ i . (v) This group G is the Galoisgroup of the equation (9) as it is the largest subgroup of the symmetric group S that leavesinvariant the basic set (10) of rational functions of the roots with coefficients in Q ( a, b ).Now we know from Newton’s relations (7) that x + x = − a and x x = b so the nextrelation x − x = ξ = √ a − b (11)is not rational in the field Q ( a, b ), but it is in the field Q ( a, b, ξ ) which is by definition theextension of Q ( a, b ) to a field of rational expressions in a, b and ξ .It should be apparent that now only the first 4 permutations in G leave these relations (10)and (11) invariant: G = (cid:8) σ = (1)(2)(3)(4) , σ = (12)(3)(4) , σ = (1)(2)(34) , σ = (12)(34) (cid:9) since x = x and x = x .Now G is a normal subgroup of G with prime index 2. Notice also that the ξ in equation(11) is in fact a solution of the partial resolvent polynomial of degree | G || G | = 2 in Q ( a, b ) ξ − a + 4 b = 0 (12)From the relations (10) we can also derive a new expression x − x = 2 ξ (13)which we can view (vi) as a root of the partial resolvent polynomial of degree 2 in Q ( a, b, ξ )2 ξ + a − ξ = 0 (14)and thus the relation (13) is rational in the extended field Q ( a, b, ξ , ξ ). Now the groupleaving invariant all previous root relations plus this one (13) is G = (cid:8) σ = (1)(2)(3)(4) , σ = (12)(3)(4) (cid:9) This group G is also normal in G with prime index 2, as expected from the degree of thepartial resolvent equation (14).Likewise, the relation x − x = 2 ξ (15)is a root of the partial resolvent polynomial equation of degree 22 ξ + a + ξ = 0 (16)and then only the identity permutation leaves all these root relations (10), (11), (13) and (15)invariant in the extension field Q ( a, b, ξ , ξ , ξ ). Thus G = (cid:8) σ = (1)(2)(3)(4) (cid:9) ≡ is trivially a normal subgroup of G with prime index 2 too.We have thus obtained a sequence of subgroups (cid:67) G (cid:67) G (cid:67) G called a composition series of the Galois group of prime indexes 2 : 2 : 2. The fact that allthe above resolvent equations are binomial equations x p − A = 0 with p prime is intimatelyconnected with the fact that the equation is solvable by radicals. The fact that all the indexesin the composition series of a particular group are prime numbers establishes the group as solvable . (v) In fact G is a normal subgroup of S of prime index | S || G | = = 3). (vi) Squaring both sides of (13), adding x and noting that x = x = ξ and ξ = x − x , we get x − x x + x + x = − a + 2 ξ + x = 4 ξ + x and thus (14) eferences [Art38] E. Artin. Foundations of Galois Theory-New York University LectureNotes . New York University, 1938. 9[Asc08] M. Aschbacher. The status of the Classification of the Finite SimpleGroups. 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