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Galois theory of equations

In: Introduction to Algebra

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  • R. Kochendörffer

    (University of Dortmund)

Abstract

As is well known the roots of a quadratic equation can be computed from its coefficients by rational operations (i.e. addition, subtraction, multiplication, and division) and the extraction of a square root. In § 7.3 we shall derive similar formulae for the roots of cubic and quartic equations. Since these formulae involve only rational operations and root extractions we say that cubic and quartic equations are soluble by radicals. After vain attempts of numerous mathematicians to solve equations of degree higher than four by radicals Ruffini and Abel proved that equations of degree five cannot be solved by radicals. Later, Galois derived a necessary and sufficient condition under which an equation is soluble by radicals. The work of Galois is of great importance for the development of mathematical concepts. In this chapter we shall apply the results of the previous chapter to the study of algebraic equations and related topics.

Suggested Citation

  • R. Kochendörffer, 1972. "Galois theory of equations," Springer Books, in: Introduction to Algebra, chapter 7, pages 243-266, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-8179-9_7
    DOI: 10.1007/978-94-009-8179-9_7
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