Algebra for beginners. From the solution of equations to Galois theory. (Algebra für Einsteiger. Von der Gleichungsauflösung zur Galois-Theorie.) (German)
[B] Wiesbaden: Vieweg. xvii, 193 S. EUR 19.90; sFr. 33.60 (2002). ISBN 3-528-03192-1/pbk
At least since Italian mathematicians found more or less the general solution of equations of third degree in the first half of the sixteenth century, the question about the solution of polynomial equations with real coefficients has been in the center of interest and development of algebra. This situation changed in the nineteenth century with the outgrowth of abstract algebra out of polynomial algebra.
The book at hand tries to explain this development to a reader who knows mathematics from high school. More difficult parts of the book that need higher mathematics in order to be understood are marked by grey lines.
The book has 10 chapters: 1. Cubic equations, 2. Casus irreducibilis -- the birth of complex numbers, 3. Biquadric equations, 4. Equations of n-th degree and their properties, 5. The search for further solution formulas, 6. Equations, whose degree may be reduced, 7. The construction of regular polygons, 8. The solution of equations of the fifth degree, 9. The Galois group of an equation, 10. Algebraic structures and Galois theory.
Many historical details add to the interest of the presented mathematical problems. Even if a mathematically unexperienced reader will not understand all details, he may be encouraged to further read mathematical texts. The book will also be useful for students of an algebra course for a better understanding of the historical development.
Helmut Koch (Berlin)
Zentralblatt MATH: Zbl 0995.12001