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Galois Theory Richard Koch September 12, 2021 Contents Preface 4 1 Arnold's Proof of the Abel-Ruffini Theorem 18 2 Finite Group Theory 25 2.1 The Extension Problem; Simple Groups.................... 25 2.2 An Isomorphism Lemma............................. 26 2.3 Jordan-Holder................................... 27 2.4 The Symmetric and Alternating Groups.................... 29 3 The Quadratic, Cubic, and Quartic Formulas 32 3.1 The Quadratic Formula............................. 32 3.2 The Cubic Formula................................ 33 3.3 The Quartic Formula............................... 38 4 Field Extensions and Root Fields 40 4.1 Motivation for Field Theory........................... 40 4.2 Fields....................................... 42 4.3 An Important Example............................. 43 4.4 Extension Fields................................. 43 4.5 Algebraic Extensions; Root Fields....................... 44 4.6 Irreducible Polynomials over Q ......................... 45 4.7 The Degree of a Field Extension........................ 47 4.8 Existence of Root Fields............................. 48 4.9 Isomorphism and Uniqueness.......................... 50 4.10 Putting It All Together............................. 50 5 Splitting Fields 52 5.1 Factoring P .................................... 52 5.2 The Splitting Field................................ 52 1 CONTENTS 2 5.3 Proof that Splitting Fields Are Unique..................... 54 5.4 Uniqueness of Splitting Fields, and P (X) = X3 − 2.............. 56 6 Finite Fields 59 6.1 Finite Fields................................... 59 7 Beginning Galois Theory 61 7.1 Motivation for Galois Theory.......................... 61 7.2 Motivation; Putting the Ideas Together.................... 64 7.3 The Galois Group................................ 65 8 Galois Extensions and the Fundamental Theorem 67 8.1 Galois Extensions................................. 67 8.2 Fundamental Theorem of Galois Theory.................... 70 8.3 Important Note.................................. 72 8.4 An Example.................................... 72 8.5 Finite Fields Again................................ 72 9 Concrete Cases of the Theory 74 9.1 Cyclotomic Fields................................. 74 9.2 Galois Group of a Radical Extension...................... 75 9.3 Structure of Extensions with Cyclic Galois Group............... 75 10 Solving Polynomials by Radicals 77 10.1 Solving Polynomial Equations with Radicals.................. 77 10.2 The Flaw..................................... 78 10.3 The Fix...................................... 79 10.4 Galois' Theorem on Solving Via Radicals................... 80 10.5 Solving Generic Equations by Radicals..................... 82 10.6 A Polynomial Equation Which Cannot Be Solved by Radicals........ 84 11 Straightedge and Compass Constructions 87 11.1 Constructions With Straightedge and Compass................ 87 11.2 Complex Extensions and Constructions.................... 91 11.3 Trisecting Angles; Doubling the Cube..................... 92 12 Irrationality and Transcendence of π and e 94 12.1 Irrationality e and π ............................... 94 12.2 Transcendence of e ................................ 96 12.3 Intermission: Fundamental Theorem of Symmetric Polynomials....... 98 12.4 Transcendence of π ............................... 99 CONTENTS 3 13 Special Cases 102 13.1 The Discriminant................................. 102 13.2 The Cubic Case.................................. 103 13.3 Cyclotomic Fields................................. 105 13.4 Galois Group of Xp − a over Q ......................... 109 14 Constructing Regular Polygons 111 14.1 Constructing Regular Polygons......................... 111 15 Normal and Separable Extensions 115 15.1 Normal Extensions, Separable Extensions, and All That........... 115 16 Galois Theory and Reduction Modulo p 120 16.1 Preview...................................... 120 16.2 X5 − X − 1.................................... 121 16.3 A Tricky Point Concerning Straightedge and Compass Constructions.... 122 16.4 Polynomials with Galois Group Sn ....................... 124 16.5 Every Finite G is a Galois Group........................ 125 16.6 Proof of Dedekind's Theorem.......................... 125 Preface The beginnings of algebra, and the discovery of the quadratic formula, are hidden in the mists of time. At first, algebra was written entirely with words: \the thing plus oneequals two." This \rhetorical algebra" was created in Babylonia and lasted until the early fifteenth century. Diophantus may have been the first person to use symbols for some of these words, in a book named Arithmetica written around 200 AD. Algebra was extensively developed by Arabic mathematicians starting around 700 AD, eventually giving the subject its name. The modern symbolic approach begins to appear in full dress in the works of Francois Viete toward the end of the 1500's and Rene Descartes' La Geometrie of 1637. Signs of quadratic equations appear early in this long development, but it is difficult to pick a particular moment that the quadratic formula appears in the precise form we use today. Perhaps it is better to say that it has been a part of the subject for many centuries. Cubic equations appear very early in this history, even in Babylonian times. By the 1500's, with the quadratic formula in its modern form, mathematicians began searching for an analogous cubic formula. The first glimpse of the formula was seen by Scipione del Ferro around 1500, but he did not publish the result, which was found in his mathematical papers after his death. The formula was rediscovered by Niccol Tartaglia in 1530, and used by him in a mathematical contest against del Ferro's son-in-law, who had discovered the formula in del Ferro's papers. Later Tartaglia revealed the formula to Gerolamo Cardano, on condition that he not reveal the formula until Tartaglia published it. Cardano generally kept his word, but he did reveal the formula to an apprentice named Lodovico Ferrari, who used it to produce a formula for solving quartic equations. This put Cardano in a bind, since Tartaglia did not publish and Ferrari could not show his solution without revealing the cubic formula. Eventually Cardano recalled the ancient contest of Tartaglia and del Ferro's son-in-law, and visited del Ferro's widow, who turned out to have kept her husband room intact since his death. There Cardano found del Ferro's solution, and published it while claiming not to break his oath to Tartaglia. All of this is covered in much greater detail in the wonderful book Cardano, the Gambling Scholar by Oystein Ore. The quadratic formula can be obtained by introducing a new variable y = x+λ and writing 4 CONTENTS 5 the quadratic in terms of y rather than x. Geometrically, this amounts to translating the graph of the quadratic by λ. If λ is chosen appropriately, the linear term in the equation for y cancels out and we obtain y2 + A = 0, which is easily solved. The same trick words for cubic equations, reducing an arbitrary cubic to the form x3 + Ax + B = 0 The cubic formula states that x is then equal to s r s r 3 B B2 A3 3 B B2 A3 x = − + + + − − + 2 4 27 2 4 27 This formula has some unexpected properties. If A and B are real, the cubic either has one real root or three real roots, except for trivial edge cases. If it has one real root, the expression under the square root is positive and then the formula gives the real root since any real number has a real cube root. But if there are three real roots, the expression under the square root is negative. This suggests that a separate formula may be required to handle the case of three real roots. In 1572, Bombelli published L'Algebra. In this book he advanced algebraic notation and added many examples from Diophantus after becoming one of the first modern Europeans to rediscover this book. But Bombelli is mainly remembered today for a specific example in his book, the solution of the cubic equation x3 − 15x − 4 = 0. He obtains x = 4, which can be easily checked. It is his method which is significant, for he obtains this root from the cubic formula. According to the formula, q p q p p p x = 3 2 + −121 + 3 2 − −121 = 3 2 + 11i + 3 2 − 11i Bombelli then notices that (2 + i)3 = 2 + 11i and (2 − i)3 = 2 − 11i and thus x = (2 + i) + (2 − i) = 4 Before this book appeared, complex numbers were not taken seriously, and it is their use in the cubic formula which gave mathematicians the first glimpse of their importance. Later mathematicians computed complex cube roots using trigonometry and then the cubic formula could solve all cubic equations. Incidentally, we will later prove that cubic equations with three real roots cannot be solved using only real radicals, so the cubic formula is the only game in town. After Ferrari found a quartic formula, mathematicians tried to find formulas for higher degree equations. In the late 1700's, this work was taken on by Lagrange (who also created the metric system during the French Revolution). Lagrange created various linear combi- nations of the roots, called resolvents, and found that each satisfied a polynomial equation CONTENTS 6 called its resolvent equation. Finding the resolvent by solving this equation often led to a complete solution of the original equation. In the quadratic, cubic, and quartic cases, resolvents exist whose resolvent equations have lower degree than the original equation, so there is a general inductive procedure to solve these equations. However, Lagrange was unable to find a general resolvent
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