Classical Galois Theory, With Examples by Lisl Gaal 3rd Ed 1979 HB Book

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PREFACE

Galois theory is one of the most beautiful subjects in mathematics, but it is hard to appreciate this fact fully without seeing specific examples. Numerous examples are therefore included throughout the text, in the hope that they will lead to a deeper understanding and genuine appreciation of the more abstract and advanced literature on Galois theory.

The book is intended for beginning graduate students who already have some background in abstract algebra, including some elementary theory of groups, rings, and fields.

Chapter I consists of results that should be used as background references of definitions, theorems, and examples. The study of Galois theory proper begins in Chapter II, which introduces automorphisms of fields and related topics. In Chapter III we define normal extensions, prove the fundamental theorem, and give examples that illustrate what happens here. Chapter IV contains applications. We prove that an equation is solvable if and only if its group is solvable, taking the meaning of "solvable" in its strict sense-that a solution can actually be produced and written down. Since the actual solution of a solvable polynomial requires that we first determine the group, we also include a procedure for doing this. The procedures that accomplish this are complicated, as one would expect of a process that must cover the solution of all possible solvable equations, but nonetheless it is an algorithm, as is the procedure for finding the group of a polynomial. Among the other