4 Chapter 1. Groups I of degree six, not to a polynomial of smaller degree. Informally, let us say that a polynomial f(x) is solvable by radicals if there is a formula for its roots which has the same form as the quadratic, cubic, and quartic formulas that is, which uses only arithmetic operations and roots of numbers involving the coeﬃcients of f(x). In 1799, Ruﬃni claimed that the general quintic formula is not solvable by radicals, but his contemporaries did not accept his proof his ideas were, in fact, correct, but his proof had gaps. In 1815, Cauchy introduced the multiplication of permutations, and he proved basic properties of what we call the symmetric group Sn for example, he introduced the cycle notation and proved unique factorization of permutations into disjoint cycles. In 1824, Abel gave an acceptable proof that there is no quintic formula in his proof, Abel constructed permutations of the roots of a quintic, using certain rational functions introduced by Lagrange. Galois, the young wizard who was killed before his 21st birthday, modified Lagrange’s rational functions but, more important, he saw that the key to understanding which polynomials are solvable by radicals involved what he called groups: subsets of the symmetric group Sn that are closed under composition—in our language, subgroups of Sn. To each polynomial f(x), he associated such a group, nowadays called the Galois group of f(x). He recognized conjugation, normal subgroups, quotient groups, and simple groups, and he proved, in our language, that a polynomial (over a field of characteristic 0) is solvable by radicals if and only if its Galois group is a solvable group (solvability being a property generalizing commutativity). A good case can be made that Galois was one of the most important founders of modern Algebra. For an excellent account of this history, we recommend the book, Tignol, Galois’ Theory of Algebraic Equations. Exercises 1.1. Given M, N ∈ C, prove that there exist g, h ∈ C with g + h = M and gh = N. ∗ 1.2. The following problem, from an old Chinese text, was solved by Ch’in Chiu-shao (Qin Jiushao) in 1247. There is a circular castle, whose diameter is unknown it is provided with four gates, and two li out of the north gate there is a large tree, which is visible from a point six li east of the south gate. What is the length of the diameter? S E N C O T 2 r r r a 6 Figure 1.1. Castle problem. Hint. The answer is a root of a cubic polynomial. 1.3. (i) Find the complex roots of f(x) = x3 − 3x + 1.

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