08-17-2026, 05:31 PM
Galois Theory Through Exercises
Author: Juliusz Brzeziński
Publication: 2018
Publisher: Springer International Publishing
Series: Springer Undergraduate Mathematics Series
Galois Theory Through Exercises takes an unusually practical route into one of the central subjects of abstract algebra. Rather than developing Galois theory primarily as a long sequence of definitions and proofs, Juliusz Brzeziński organizes much of the learning around nearly 500 exercises, ranging from relatively straightforward examples to substantially more demanding problems. The book begins with algebraic equations and develops the machinery needed for Galois theory through field extensions, irreducible polynomials, algebraic and splitting fields, automorphism groups, normal and separable extensions, and finally Galois extensions. The exercises are not simply supplementary problems: they form an important part of the mathematical development, encouraging the reader to discover how the abstract theory works in concrete situations.
The later chapters move toward some of the most interesting consequences of the theory: cyclotomic extensions, solvable groups, solvability of polynomial equations by radicals, geometric constructions and methods for actually computing Galois groups. Brzeziński also goes beyond the most standard undergraduate treatment, touching on topics such as Tschirnhausen transformations, Galois resolvents, Nagell's proof concerning quintic equations, the inverse Galois problem and cyclotomic fields. Computer-assisted exercises are included as well, principally using Maple, although reviewers note that alternatives such as Sage or PARI can often be used. Proofs, historical remarks, hints, answers and selected solutions complement the exercises, so the book can function both as a course text and as a resource for independent study.
The book's greatest strength is therefore its learning-by-doing philosophy. Galois theory can initially seem forbiddingly abstract because several layers of algebra—groups, fields, polynomial rings and field automorphisms—interact simultaneously. Brzeziński tries to make these structures tangible by repeatedly applying them to explicit equations and extensions. This makes the book particularly attractive to mathematically mature undergraduate students who already know some abstract algebra and want to move from merely reading Galois theory to actually working with it. It is also a valuable source of problems for lecturers. A Mathematical Association of America review specifically praised the book's attempt to make abstract algebra more concrete and judged the approach successful.
Key Takeaways
Springer — Galois Theory Through Exercises
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[url=https://www.goodreads.com/book/show/41076991-galois-theory-through-exercises?utm_source=chatgpt.com]Goodreads — Galois Theory Through Exercises
Author: Juliusz Brzeziński
Publication: 2018
Publisher: Springer International Publishing
Series: Springer Undergraduate Mathematics Series
Galois Theory Through Exercises takes an unusually practical route into one of the central subjects of abstract algebra. Rather than developing Galois theory primarily as a long sequence of definitions and proofs, Juliusz Brzeziński organizes much of the learning around nearly 500 exercises, ranging from relatively straightforward examples to substantially more demanding problems. The book begins with algebraic equations and develops the machinery needed for Galois theory through field extensions, irreducible polynomials, algebraic and splitting fields, automorphism groups, normal and separable extensions, and finally Galois extensions. The exercises are not simply supplementary problems: they form an important part of the mathematical development, encouraging the reader to discover how the abstract theory works in concrete situations.
The later chapters move toward some of the most interesting consequences of the theory: cyclotomic extensions, solvable groups, solvability of polynomial equations by radicals, geometric constructions and methods for actually computing Galois groups. Brzeziński also goes beyond the most standard undergraduate treatment, touching on topics such as Tschirnhausen transformations, Galois resolvents, Nagell's proof concerning quintic equations, the inverse Galois problem and cyclotomic fields. Computer-assisted exercises are included as well, principally using Maple, although reviewers note that alternatives such as Sage or PARI can often be used. Proofs, historical remarks, hints, answers and selected solutions complement the exercises, so the book can function both as a course text and as a resource for independent study.
The book's greatest strength is therefore its learning-by-doing philosophy. Galois theory can initially seem forbiddingly abstract because several layers of algebra—groups, fields, polynomial rings and field automorphisms—interact simultaneously. Brzeziński tries to make these structures tangible by repeatedly applying them to explicit equations and extensions. This makes the book particularly attractive to mathematically mature undergraduate students who already know some abstract algebra and want to move from merely reading Galois theory to actually working with it. It is also a valuable source of problems for lecturers. A Mathematical Association of America review specifically praised the book's attempt to make abstract algebra more concrete and judged the approach successful.
Key Takeaways
- Exercise-driven approach: Much of Galois theory is learned by solving problems rather than passively following exposition.
- Broad classical coverage: It develops the path from polynomial equations and field extensions through Galois groups and solvability by radicals.
- More than a standard textbook: Historical material, computational exercises, geometric constructions and topics connected with the inverse Galois problem broaden the treatment.
- Excellent for active self-study: Hints, answers and selected solutions make it especially useful for readers who want to develop genuine problem-solving ability in abstract algebra.
Springer — Galois Theory Through Exercises
[/url]
[url=https://www.goodreads.com/book/show/41076991-galois-theory-through-exercises?utm_source=chatgpt.com]Goodreads — Galois Theory Through Exercises
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