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Abstract algebra and famous impossibilities. (English) Zbl 0758.12001

Universitext. New York etc.: Springer-Verlag (ISBN 0-387-97661-2/pbk). x, 187 p. (1991).
The purpose of this book is to provide complete (negative) answers to the following three classical problems : duplicating the cube, trisecting an angle, squaring the circle. The level is kept as elementary as possible, the most difficult part being, of course, the proof of the transcendence of \(\pi\). Few prerequisites are needed : essentially only very basic facts from linear algebra.
After an extremely brief historical introduction, the authors develop in the first four chapters the necessary algebraic tools : fields, rings, vector spaces, polynomials, algebraic numbers, field extensions. In Chapter 5 they define carefully the rules of constructions with straightedge and compass in order to connect the geometry of such constructions with the previous algebraic machinery. The proof of Wantzel’s result (negative answer to the first two problems) is given in Chapter 6. The impossibility of the quadrature of the circle requires Lindemann’s result on the transcendence of \(\pi\) whose proof is given in Chapter 7. The authors conclude by a short discussion of three further related problems : construction of regular polygons, solving quintic equations, integrating functions in closed form. Each chapter contains exercises and ends with a valuable list of references for additional reading.

MSC:

12-02 Research exposition (monographs, survey articles) pertaining to field theory
51-02 Research exposition (monographs, survey articles) pertaining to geometry
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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