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2000 Mathematics Subject Classification: 12E10, 12E12, 12F10. Key words and phrases: solvable quintic equations, Watson’s method. The second and third authors were supported by research grants from the Natural Sciences and Engineering Research Council of Canada.
WATSON’S METHOD OF SOLVING A QUINTIC EQUATION
Melisa J. Lavallee, Blair K. Spearman, and Kenneth S. Williams
Abstract. Watson’s method for determining the roots of a solvable quintic equation in radical form is examined in complete detail. New methods in the spirit of Watson are constructed to cover those exceptional cases to which Watson’s original method does not apply, thereby making Watson’s method completely general. Examples illustrating the various cases that arise are presented.
As f(x) is solvable and irreducible, we have [4, p. 390]
(2.4)
δ > 0.
We set (2.5)
K = E + 3C2,
(2.6)
L = −2DF + 3E2 − 2C2E + 8CD2 + 15C4,
(2.7)
M = CF 2 − 2DEF + E3 − 2C2DF − 11C2E2
see for example [5, p. 987].
4 MELISA J. LAVALLEE, BLAIR K. SPEARMAN, AND KENNETH S. WILLIAMS
If θ = 0, ±C Watson’s method of determining the roots of f(x) = 0 in radical form is given in the next theorem.
are the roots of
√
(2.9)
g(x) = x6 − 100Kx4 + 2000Lx2 − 32 δx + 40000M ∈ Q[x].
WATSON’S METHOD
3
Watson [1] has observ√ed as f(x) is solvable and irredu√cible that g(x) has a
Theorem 1. Let f(x) be the solvable irreducible quintic polynomial (2.1). Suppose that θ = 0, ±C. Set
(2.17) p(T ) = T 4 + (−14Cθ2 − 2D2 + 2CE − 2C3)T 2 + 16Dθ3T + (−25θ6 + (35C2 + 6E)θ4 + (−11C4 − 2CD2 − 4C2E − E2)θ2 + (C6 + 2C3D2 − 2CD2E − 2C4E + C2E2 + D4)
then the five roots of f(x) = 0 are
(2.16)
x = ωu1 + ω2u2 + ω3u3 + ω4u4,
where ω runs through the fifth roots of unity.
Proof. This follows from the identity ωu1 + ω2u2 + ω3u3 + ω4u4 5 − 5U ωu1 + ω2u2 + ω3u3 + ω4u4 3 − 5V (ωu1 + ω2u2 + ω3u3 + ω4u4)2 + 5W (ωu1 + ω2u2 + ω3u3 + ω4u4) + 5(X − Y ) − Z = 0,
+28CD2E − 16D4 + 35C4E − 40C3D2 − 25C6. Let x1, x2, x3, x4, x5 ∈ C be the five roots of f (x). Cayley [2] has shown that
(2.8)
φ1 = x1x2 + x2x3 + x3x4 + x4x5 + x5x1 − x1x3 − x3x5 − x5x2 − x2x4 − x4x1, φ2 = x1x3 + x3x4 + x4x2 + x2x5 + x5x1 − x1x4 − x4x5 − x5x3 − x3x2 − x2x1, φ3 = x1x4 + x4x2 + x2x3 + x3x5 + x5x1 − x1x2 − x2x5 − x5x4 − x4x3 − x3x1, φ4 = x1x2 + x2x5 + x5x3 + x3x4 + x4x1 − x1x5 − x5x4 − x4x2 − x2x3 − x3x1, φ5 = x1x3 + x3x5 + x5x4 + x4x2 + x2x1 − x1x5 − x5x2 − x2x3 − x3x4 − x4x1, φ6 = x1x4 + x4x5 + x5x2 + x2x3 + x3x1 − x1x5 − x5x3 − x3x4 − x4x2 − x2x1,
(2.12)
u1u4 + u2u3 = −2C,
(2.13)
u1u22 + u2u24 + u3u21 + u4u23 = −2D,
(2.14)
u21u24 + u22u23 − u31u2 − u32u4 − u33u1 − u34u3 − u1u2u3u4 = E,
(2.15) u51 + u52 + u53 + u54 − 5(u1u4 − u2u3)(u21u3 − u22u1 − u23u4 + u24u2) = −F,
1
2 MELISA J. LAVALLEE, BLAIR K. SPEARMAN, AND KENNETH S. WILLIAMS
2. Watson’s method. Let f(x) be a monic solvable irreducible quintic polynomial in Q[x]. By means of a linear change of variable we may suppose that the coefficient of x4 is 0 so that
1. Introduction. In the 1930’s the English mathematician George Neville Watson (1886-1965) devoted considerable effort to the evaluation of singular moduli and class invariants arising in the theory of elliptic functions [6]-[11]. These evaluations were given in terms of the roots of polynomial equations whose roots are expressible in terms of radicals. In order to solve those equations of degree 5, Watson developed a method of finding the roots of a solvable quintic equation in radical form. He described his method in a lecture given at Cambridge University in 1948. A commentary on this lecture was given recently by Berndt, Spearman and Williams [1]. This commentary included a general description of Watson’s method. However it was not noted by Watson (nor in [1]) that there are solvable quintic equations to which his method does not apply. In this paper we describe Watson’s method in complete detail treating the exceptional cases separately, thus making Watson’s method applicable to any solvable quintic equation. Several examples illustrating Watson’s method are given. Another method of solving the quintic has been given by Dummit [4].