All rights are reserved Translated by: Diab Publishing & Distribution www.Diab-publishing.com Preface This book contains valuable lessons of scientific value in mathematics, especially in relation to the wide field of sports (Nouh and Galois). This book is the unique encyclopedia of mathematics that helps in the teaching profession and in the culture of mathematics and in the contemporary aspects of this book with the various sciences of physics, engineering and technical sciences. The researcher Mr. Nouh Hassan has relied on special curricula for educational education, research education and mathematics. It is a riches for the pioneers of mathematics and the world of the comprehensive scientific level. Author Nouh Hassan 9/12/2019 Mr. Nouh Hassan in brief: Born in 1957, Maarake Village, Tyre in southern Lebanon. He received his primary and intermediate education at the official battle school and finished high school at the official secondary school. He graduated from the Teachers' and Teachers' College in Nabatieh in 1979. He obtained his educational leave from the Lebanese University in mathematics in 1985. He was ranked first in Mathematics in the Lebanese Civil Service Council in 1995. He completed his graduation from the Faculty of Education at the Lebanese University in 1996 with the rank of secondary professor in the staff of the Ministry of National Education. Participated in many educational activities, training courses and public events. Contents Historical Introduction……………………………..……………….…….. 1 1. Classical Algebra….…………………………………………….……….17 2. The Fundamental Theorem of Algebra……………….…....……….... 35 3. Factorisation of Polynomials………………………...……..……….…. 47 4. Field Extensions……………………………………………….…….….. 63 5. Simple Extensions……………………………………………….….….. 71 6. The Degree of an Extension………………………...…………..…….. 79 7. Ruler-and-Compass Constructions……………………………..……...87 8. The Idea Behind Galois Theory……………………………….…….. 107 9. Normality and Separability………………………..….……….……… 129 10. Counting Principles………………………………………...…………. 137 11. Field Automorphisms…………………………………....……………. 145 12. The Galois Correspondence………………………………….……… 151 13. A Worked Example……………………………………....…………… 155 14. Solubility and Simplicity……………………………….……………… 161 15. Solution by Radicals…………………………………..………………. 171 16. Abstract Rings and Fields…………………...………..……………… 181 17. Abstract Field Extensions………………………….…………………. 193 18. The General Polynomial Equation………………..…………………. 205 19. Finite Fields………………………………………….………………….221 20. Regular Polygons……………………………………………...……… 227 21. Circle Division……………………………………………………..……243 22. Calculating Galois Groups…………………………………………… 267 23. Algebraically Closed Fields…………………………………......…… 277 24. Transcendental Numbers……………………………………...…….. 285 25. What Did Galois Do or Know?....................................................... 295 References……………………………………………………...……… 309 Historical Introduction Mathematics has a rich history, going back at least 5000 years. Very few subjects still make use of ideas that are as old as that, but in mathematics, important dis- coveries have lasting value. Most of the latest mathematical research makes use of theorems that were published last year, but it may also use results first discovered by Archimedes, or by some unknown Babylonian mathematician, astronomer, or priest. For example, ever since Archimedes proved (around 250 BC) that the volume of a 4 3 sphere is what we would now write as 3 pr , that discovery has been available to any mathematician who is aware of the result, and whose research involves spheres. Al- though there are revolutions in mathematics, they are usually changes of viewpoint or philosophy; earlier results do not change—although the hypotheses needed to prove them may. In fact, there is a word in mathematics for previous results that are later changed: they are called ‘mistakes’. The history of Galois theory is unusually interesting. It certainly goes back to 1600 BC, where among the mud-brick buildings of exotic Babylon, some priest or mathematician worked out how to solve a quadratic equation, and they or their stu- dent inscribed it in cuneiform on a clay tablet. Some such tablets survive to this day, along with others ranging from tax accounts to observations of the motion of the planet Jupiter, Figure 1 (Left). Adding to this rich historical brew, the problems that Galois theory solves, pos- itively or negatively, have an intrinsic fascination—squaring the circle, duplicating the cube, trisecting the angle, constructing the regular 17-sided polygon, solving the quintic equation. If the hairs on your neck do not prickle at the very mention of these age-old puzzles, you need to have your mathematical sensitivities sharpened. If those were not enough: Galois himself was a colourful and tragic figure—a youthful genius, one of the thirty or so greatest mathematicians who have ever lived, but also a political revolutionary during one of the most turbulent periods in the history of France. At the age of 20 he was killed in a duel, ostensibly over a woman and quite possibly with a close friend, and his work was virtually lost to the world. Only some smart thinking by Joseph Liouville, probably encouraged by Galois’s brother Alfred, rescued it. Galois’s story is one of the most memorable among the lives of the great mathematicians, even when the more excessive exaggerations and myths are excised. Our tale therefore has two heroes: a mathematical one, the humble polynomial equation, and a human one, the tragic genius. We take them in turn. 1 2 Historical Introduction FIGURE 1: Left: A Babylonian clay tablet recording the motion of Jupiter. Right:A page from Pacioli’s Summa di Arithmetica. Polynomial Equations A Babylonian clay tablet from about 1600 BC poses arithmetical problems that reduce to the solution of quadratic equations (Midonick 1965 page 48). The tablet also provides firm evidence that the Babylonians possessed general meth- ods for solving quadratics, although they had no algebraic notation with which to express their solution. Babylonian notation for numbers was in base 60, so that (when transcribed into modern form) the symbols 7,4;3,11 denote the number 2 −1 −2 191 7 × 60 + 4 × 60 + 3 × 60 + 11 × 60 = 25440 3600 . In 1930 the historian of sci- ence Otto Neugebauer announced that some of the most ancient Babylonian problem tablets contained methods for solving quadratics. For instance, one tablet contains this problem: find the side of a square given that the area minus the side is 14,30. Bearing in mind that 14;30 = 870 in decimal notation, we can formulate this prob- lem as the quadratic equation x2 − x = 870 The Babylonian solution reads: Take half of 1, which is 0;30, and multiply 0;30 by 0;30, which is 0;15. Add this to 14,30 to get 14,30;15. This is the square of 29;30. Now add 0;30 to 29;30. The result is 30, the side of the square. Polynomial Equations 3 Although this description applies to one specific equation, it is laid out so that similar reasoning can be applied in greater generality, and this was clearly the Babylonian scribe’s intention. The method is the familiar procedure of completing the square, which nowadays leads to the usual formula for the solution of a quadratic. See Joseph (2000) for more on Babylonian mathematics. The ancient Greeks in effect solved quadratics by geometric constructions, but there is no sign of an algebraic formulation until at least AD 100 (Bourbaki 1969 page 92). The Greeks also possessed methods for solving cubic equations, which involved the points of intersection of conics. Again, algebraic solutions of the cubic were unknown, and in 1494 Luca Pacioli ended his Summa di Arithmetica (Figure 1, right) with the remark that (in his archaic notation) the solution of the equations x3 + mx = n and x3 + n = mx was as impossible at the existing state of knowledge as squaring the circle. This state of ignorance was soon to change as new knowledge from the Middle and Far East swept across Europe and the Christian Church’s stranglehold on in- tellectual innovation began to weaken. The Renaissance mathematicians at Bologna discovered that the solution of the cubic can be reduced to that of three basic types: x3 + px = q;x3 = px + q, and x3 + q = px. They were forced to distinguish these cases because they did not recognise the existence of negative numbers. It is thought, on good authority (Bortolotti 1925), that Scipio del Ferro solved all three types; he certainly passed on his method for one type to a student, Antonio Fior. News of the solution leaked out, and others were encouraged to try their hand. Solutions for the cubic equation were rediscovered by Niccolo Fontana (nicknamed Tartaglia, ‘The Stammerer’; Figure 2, left) in 1535. One of the more charming customs of the period was the public mathemati- cal contest, in which mathematicians engaged in mental duels using computational expertise as their weapons. Mathematics was a kind of performance art. Fontana demonstrated his methods in a public competition with Fior, but refused to reveal the details. Finally he was persuaded to tell them to the physician Girolamo Cardano, having first sworn him to secrecy. Cardano, the ‘gambling scholar’, was a mixture of genius and rogue, and when his Ars Magna (Figure 2, right) appeared in 1545, it contained a complete discussion of Fontana’s solution. Although Cardano claimed motives of the highest order (see the modern translation of his The Book of My Life, 1931), and fully acknowledged Fontana as the discoverer, Fontana was justifiably annoyed. In the ensuing wrangle, the history of the discovery became public knowl- edge. The Ars Magna also contained a method, due to Ludovico Ferrari, for solving the quartic equation by reducing it to a cubic. Ferrari was one of Cardano’s students, so presumably he had given permission for his work to be published::: or perhaps a student’s permission was not needed. All the formulas discovered had one striking property, which can be illustrated by Fontana’s solution x3 + px = q : s r s r 3 q p3 q2 3 q p3 q2 x = + + + − + 2 27 4 2 27 4 4 Historical Introduction FIGURE 2: Left: Niccolo Fontana (Tartaglia), who discovered how to solve cubic equations.