Chapter 1 Groups I Section 1.1. Classical Formulas As Europe emerged from the Dark Ages, a major open problem in Mathematics was finding roots of polynomials. The quadratic formula had been known for over two thousand years and, arising from a tradition of public mathematical contests in Venice and Tuscany, formulas for the roots of cubics and quartics had been found in the early 1500s. Consider the cubic F (X) = X3 + bX2 + cX + d.1 The change of variable X = x − 1 3 b yields a simpler polynomial f(x) = x3 + qx + r whose roots give the roots of F (x): if u is a root of f(x), then u − 1 3 b is a root of F (x). Special cases of the cubic formula were discovered by Scipio del Ferro around 1515, and the remaining cases were completed by Niccol` o Fontana (Tartaglia) in 1535 and by Girolamo Cardano in 1545. The roots of f(x) are g + h, ωg + ω2h, and ω2g + ωh, where g3 = 1 2 ( −r + √ R ) , h = −q/3g, R = r2 + 4 27 q3, and ω = − 1 2 + i √ 3 2 is a primitive cube root of unity. This formula is derived as follows. If u is a root of f(x) = x3 + qx + r, write u = g + h, and substitute: 0 = f(u) = f(g + h) = g3 + h3 + (3gh + q)u + r. (1) Now the quadratic formula can be rephrased to say, given any pair of numbers M and N, that there are (possibly complex) numbers g and h with g + h = M and 1We must mention that modern notation was not introduced until the late 1500s, and was generally agreed upon only after the influential book of Descartes in 1637. For example, letters for variables were invented (by Vi` ete) in 1591, the equality sign = was invented (by Recorde) in 1557, and exponents were invented (by Hume) in 1636. The symbols + and − were introduced (by Widman) in 1486 (see Cajori, A History of Mathematical Notation). 1

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