An introduction to the modern theory of equations, by Florian Cajori.

186 THEORY OF EQUATIONS Since G4(4)III is an intransitive group, the quartic can be factored in the domain ~'(?,,T,. The two quadratic equations thereby obtained have as Galois groups 1, (ab), and 1, (ccl), respectively. From VI, ~ 62, we see that 2'(~s;, v= v '(vw, v,). Hence it is not necessary to adjoin more than one of the two irrationals \/u, \/w. The quartic offers a better exhibit of the Galois theory than did the quadratic and cubic equations, because not only may we select a great variety of different functions Mi at each adjunction, but we may select different groups. In the above solution the series of groups taken is G24,,, G,4), G4(4)III, G = (1, (ab)), G = 1, but another series may be chosen, viz. G24(4), G12(4), G4(4)II, G(4), 1. In Exs. 1 and 3, ~ 163, a solution of the quartic is outlined, in which this series of groups is used. Again, we may effect a solution by first adjoining a function that belongs to the cyclic group G4(4)I; say, y = aci1 4 2 2 - 2- + (t2 + a(t,2. To be sure, the first resolvent equation q (y) = 0 will be of the sixth degree, but it can be treated as an equation of the third degree and a quadratic. The number of different solutions of cubic and quartic equations which have been given since the time of Tartaglia and Cardan is enormous. For information on different solutions consult L. Matthiessen, Grundzilge der Antiken u. Modernen Algebra. It would seem that the above mode of procedure should lead to solutions of the general quintic equation. But an unexpected difficulty arises in our inability to solve all the resolvent equations. There arise resolvents of higher than the fourth degree. The Galois theory will furnish proof that the solution by radicals of the general quintic and of general equations of higher degrees is not possible. In the remaining chapters we shall demonstrate this impossibility and discuss the theory of special types of equations of higher degree which can be solved algebraically.

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 170
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2025.
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