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Groups and Galois Theory Groups and Galois Theory Course Notes Alberto Elduque Departamento de Matem´aticas Universidad de Zaragoza 50009 Zaragoza, Spain c 2009-2019 Alberto Elduque Contents Syllabus v What is this course about? vii 1 Groups 1 x 1. Definitions and examples . 1 x 2. Cyclic groups. Generators . 3 x 3. Homomorphisms . 5 x 4. Symmetric group . 9 x 5. Group actions . 13 x 6. Conjugation. Translation . 17 6.1. Action by conjugation . 17 6.2. Action by translation . 19 x 7. Sylow's Theorems . 20 x 8. Direct products . 22 x 9. Finitely generated abelian groups . 24 x 10. Solvable groups . 27 x 11. Simple groups . 30 Exercises . 32 2 Galois Theory 37 x 1. Algebraic extensions . 37 x 2. Splitting fields. Algebraic closure . 42 x 3. Separable extensions . 47 x 4. Galois group . 49 x 5. The Fundamental Theorem of Galois Theory . 55 x 6. Finite fields . 58 x 7. Primitive elements . 59 x 8. Ruler and compass constructions . 60 x 9. Galois groups of polynomials . 62 x 10. Solvability by radicals . 69 Exercises . 75 Previous exams 79 iii Syllabus This is a required course for Math Majors at the University of Zaragoza (Science School). It consists of 6 credits. It is a sequel of (and relies heavily on) the course Algebraic Structures. Lecturer: Alberto Elduque Office: Math. Building. Second floor. Algebra Section. Office no. 2. e-mail: [email protected] http://www.unizar.es/matematicas/algebra/elduque Office hours: Tuesday and Friday, from 17:00 to 20:00. Course description: The goal of the course is to understand the basic properties of the groups, which form the basic algebraic structure to under- stand and measure the symmetry, and then to proceed to study the symme- try of the algebraic equations (Galois Theory). The peak of the course will be the proof of the impossibility to solve by radicals the algebraic equations of degree ≥ 5. Exam, exercises: There will be a final exam, which will consist of some questions of a theoretical nature and some exercises. Students will be re- quired to solve and explain to their classmates some of the exercises in the notes. This will constitute a ten percent of the final mark. References: There are many good references on Abstract Algebra, which include chapters devoted to Groups and to Galois Theory. Here are just a few of them: - D.S. Dummit and R.M. Foote: Abstract Algebra (2nd edition). John Wiley and Sons, 1999. - J.J. Rotman: A First Course in Abstract Algebra (2nd edition), Pren- tice Hall, 2000. - R.B. Ash: Abstract Algebra. The Basic Graduate Year http://www.math.uiuc.edu/ r-ash/Algebra.html v vi SYLLABUS Some more specific textbooks are the following: - I. Stewart: Galois Theory. 4th ed., CRC Press, 2015. - J.P. Tignol: Galois' Theory of Algebraic Equations, World Scientific, 2001. - J.S. Milne: Group Theory and Fields and Galois Theory, http://www.jmilne.org/math/CourseNotes/index.html What is this course about? Symmetries of an equilateral triangle Fix an equilateral triangle and consider the set 2 G = fisometries of R fixing the triangleg G consists of three reflections relative to the lines that pass through a vertex and the middle point of the opposite side (see Figure 1) together with the rotations of 0, 120 and 240 degrees. Thus G consists of 6 elements: jGj = 6. Each isometry fixing the triangle permutes the vertices. The three reflec- tions correspond to the transposition of two vertices. The rotations of 120 and 240 degrees correspond to the two cyclic permutations of the vertices. σ1 . .... ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .. ... ... .. .... ... ... .... ...... ...... ... ... ... ... ... .... .... ... ... ... ... ... ... ..... ..... ... ... .. .. ... ... ........... ... ... .. .. ... ... ...... ...... ... ... ... ... ....... ....... ... ... ... ... ...... ...... ... ... ... ... ...... ...... ... ..... ..... ....... .................................................................................................................................................................................................................................................................. ....... ....... σ3 σ2 Figure 1: Symmetries of an equilateral triangle. For us, the important property of this set G is that its elements can be composed (they are bijective maps!), and the composition of any two of its elements again belongs to G. Thus, for example, σ1σ2 is the clockwise rotation of 120 degrees. These sets, endowed with a binary operation satisfying the usual prop- erties of the composition of bijective maps, are called groups. vii viii WHAT IS THIS COURSE ABOUT? Solutions of equations by radicals We are quite used to find the solutions of the degree 2 equations: ax2 + bx + c = 0; which are given by the familiar formula: p −b ± b2 − 4ac x = : 2a In the course Algebraic Structures we learned how to solve the equations of degree 3 and 4, which involve the use of square and cubic roots. The clue to deal with equations of degree ≥ 5 is to study the symme- tries of the equation. These form a group, called the Galois group of the equation. Many properties of the equation, like its solvability by radicals, are determined by the structure of its Galois group. The theory devoted to the study of the algebraic equations and their Galois groups is called Galois Theory. This course is devoted to study the basic properties of groups and of the Galois Theory of algebraic equations. Chapter 1 Groups The purpose of this chapter is the study of those sets which, like the set of isometries preserving an equilateral triangle, are endowed with a binary operation satisfying the usual properties. x 1. Definitions and examples 1.1 Definition. A group is a set G endowed with a binary operation (usually called multiplication) G × G −! G (x; y) 7! xy; which is associative ((xy)z = x(yz) for any x; y; z 2 G), there is a neutral element (which will usually be denoted by e: ex = x = xe for any x 2 G), and such that for any element x 2 G there exists an element, denoted by x−1 and called the inverse of x, such that xx−1 = x−1x = e. Moreover, if the multiplication is commutative (xy = yx for any x; y 2 G), then the group is said to be commutative or abelian. 1.2 Examples. (i) Any ring R is an abelian group with the operation given by its addition. In particular we get the abelian groups (Z; +) or (Z=nZ; +). (ii) If R is a unital ring, the set of its units R× = fr 2 R : 9s 2 R such that rs = sr = 1g is a group. The operation is given by the product of the ring. (iii) Classical groups: For any natural number n 2 N we have the fol- lowing groups of matrices (the operation is the usual product): 1 2 CHAPTER 1. GROUPS • GL(n; R) = fA 2 Matn(R) : det A 6= 0g (general linear group), • SL(n; R) = fA 2 Matn(R) : det A = 1g (special linear group), 0 • O(n; R) = fA 2 Matn(R): AA = Ing (orthogonal group), • SO(n; R) = O(n; R) \ SL(n; R) (special orthogonal group). (iv) Let Ω be an arbitrary set, then the set S(Ω) = ff :Ω ! Ω: f is a bijective mapg is a group under the composition of maps. It is called the group of permutations of Ω. The groups contained in S(Ω) are called groups of transformations of Ω. The famous Erlangen Program (Felix Klein, 1872) proposed that Geome- try is the study of invariants under the action of a group. Thus the euclidean geometry becomes the study of invariants under the isometry group (angles, distances, ..., are invarians), the affine (or projective) geometry becomes the study of invariants under the affine (or projective) group (for instance, the barycenter is an invariant), the topology becomes the study of invariants under the action of the group of homeomorphisms (compactness, connect- edness, fundamental groups, ..., are invariants), ... 1.3 Remark. Thanks to the associativity of the product, there is no need to put parentheses in products x1x2 ··· xn (n > 2). Also, we may define recursively the powers of an element of a group G as follows: x0 = e, n+1 n −n n −1 x = x x, x = (x ) , for any n 2 N. The associativity of the product n m n+m immediately gives the property x x = x for any x 2 G and n; m 2 Z. The associativity also implies the uniqueness of the inverse: If xy = e = zx, then z = ze = z(xy) = (zx)y = ey = y: As (xy)(y−1x−1) = e, it follows that (xy)−1 = y−1x−1 . 1.4 Definition. Let G be a group and let H be a nonempty subset of G. H is said to be a subgroup of G if for any x; y 2 H, both xy and x−1 belong to H. (It will be denoted by H ≤ G.) 1.5 `Silly' properties. Let H be a nonempty subset of the group G. • H ≤ G =) e 2 H. • H ≤ G () 8x; y 2 H; xy−1 2 H. • H is a subgroup of G if and only if H is closed under the multiplication in G, and it is a group with this multiplication. x 2. CYCLIC GROUPS. GENERATORS 3 1.6 Examples. • SO(n; R) ≤ SL(n; R) ≤ GL(n; R), O(n; R) ≤ GL(n; R). × • Let H be the ring of Hamilton quaternions, then H = fa+bi+cj+dk : a2 + b2 + c2 + d2 6= 0g. Consider the subset Q = {±1; ±i; ±j; ±kg. × Then Q is a subgroup of H consisting of 8 elements. It is called the quaternion group. × Another subgroup of H is given by the three-dimensional sphere: S3 = fa + bi + cj + dk : a2 + b2 + c2 + d2 = 1g: • Any group G contains two trivial subgroups: feg and G. Any other subgroup is said to be a proper subgroup of G. By abuse of notation, the trivial subgroup feg is usually denoted simply by 1 (or 0 if additive notation is being used).
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