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Mathematics of Computation

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Solving solvable quintics

Author: D. S. Dummit
Journal: Math. Comp. 57 (1991), 387-401
MSC: Primary 12E12; Secondary 12F10
Corrigendum: Math. Comp. 59 (1992), 309.
Corrigendum: Math. Comp. 59 (1992), 309.
MathSciNet review: 1079014
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Abstract: Let $ f(x) = {x^5} + p{x^3} + q{x^2} + rx + s$ be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if $ f(x)$ is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group $ {F_{20}}$ of order 20 in the symmetric group $ {S_5}$). When $ f(x)$ is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order.

References [Enhancements On Off] (What's this?)

  • [1] D. S. Dummit and R. M. Foote, Abstract algebra, Prentice-Hall, New York, 1991. MR 1138725 (92k:00007)
  • [2] I. M. Isaacs, Solution of polynomials by real radicals, Amer. Math. Monthly 92 (1985), 571-575. MR 812099 (87d:12006)
  • [3] R. Schoof and L. Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 929552 (89h:11067b)
  • [4] H. Weber, Lehrbuch der Algebra. I, Chelsea, New York, 1961.

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Article copyright: © Copyright 1991 American Mathematical Society

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