Algebraic Equations: An Introduction to the Theories of Lagrange and Galois (Dover Books on Mathematics)

Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical examples throughout the book, which appear with complete solutions.
A third of the text focuses on the basic ideas of algebraic theory, giving detailed explanations of integral functions, permutations, and groups. in addition to a very clear exposition of the symmetric group and its functions. A study of the theory of Lagrange follows. Using Lagrange's solvent as a basis for the solution of the general quadratic, cubic, and biquadratic equations. After a discussion of various groups (including isomorphic, transitive, and Abelian groups), a detailed study of Galois theory covers the properties of the Galoisian function, resolvent. and group, the general equation, reductions of the group, natural irrationality. and other features. The book concludes with the application of Galoisian theory to the solution of such special equations as Abelian, cyclic, metacyclic, and quintic equations.