Galois Theory Through Exercises
Springer International Publishing (Verlag)
978-3-319-72325-9 (ISBN)
In addition to covering standard material, the book explores topics related to classical problems such as Galois' theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading.
A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.
Juliusz Brzezinski is Professor Emeritus at the Department of Mathematical Sciences, which is a part of the University of Gothenburg and the Chalmers University of Technology, Sweden. His research concentrates on interactions between number theory, algebra and geometry of orders in algebras over global fields, in particular, in quaternion algebras. He is also interested in experimental number theory.
1 Solving algebraic equations.- 2 Field extensions.- 3 Polynomials and irreducibility.- 4 Algebraic extensions.- 5 Splitting fields.- 6 Automorphism groups of fields.- 7 Normal extensions.- 8 Separable extensions.- 9 Galois extensions.- 10 Cyclotomic extensions.- 11 Galois modules.- 12 Solvable groups.- 13 Solvability of equations.- 14 Geometric constructions.- 15 Computing Galois groups.- 16 Supplementary problems.- 17 Proofs of the theorems.- 18 Hints and answers.- 19 Examples and selected solutions.- Appendix: Groups, rings and fields.- References.- List of notations.- Index.
"This book contains a collection of exercises in Galois theory. ... The book provides the readers with a solid exercise-based introduction to classical Galois theory; it will be useful for self-study or for supporting a lecture course." (Franz Lemmermeyer, zbMATH 1396.12001, 2018)
| Erscheinungsdatum | 16.03.2018 |
|---|---|
| Reihe/Serie | Springer Undergraduate Mathematics Series |
| Zusatzinfo | XVII, 293 p. 12 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 476 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Schlagworte | algebraic field extensions • automorphism groups of fields • cubic and quartic equations • Cyclotomic Fields • field extensions • Finite Fields • Galois extensions • Galois groups of finite field extensions • Galois modules • Galois resolvents • Galois Theory • Galois theory computer-assisted examples • Galois theory exercises • inverse Galois problem • lunes of Hippocrates • Solvability of equations • solving algebraic equations of low degrees • zeros of polynomials |
| ISBN-10 | 3-319-72325-1 / 3319723251 |
| ISBN-13 | 978-3-319-72325-9 / 9783319723259 |
| Zustand | Neuware |
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