Galois Theory

5. Auflage 2022
ISBN: 978-1-032-10159-0
Verlag: Chapman and Hall/CRC

Buch, Englisch, 371 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 743 g

Since 1973, Galois theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students.

New to the Fifth Edition

- Reorganised and revised Chapters 7 and 13

- New exercises and examples

- Expanded, updated references

- Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel

- A new final chapter discussing other directions in which Galois theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations

This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.

Zielgruppe

Undergraduate Core

Autoren/Hrsg.

Stewart, Ian

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

1. Classical Algebra. 1.1. Complex Numbers. 1.2. Subfields and Subrings of the Complex Numbers. 1.3. Solving Equations. 1.4. Solution by Radicals. 2. The Fundamental Theorem of Algebra. 2.1. Polynomials. 2.2. Fundamental Theorem of Algebra. 2.3. Implications 3. Factorisation of Polynomials. 3.1. The Euclidean Algorithm. 3.2 Irreducibility. 3.3. Gauss’s Lemma. 3.4. Eisenstein’s Criterion. 3.5. Reduction Modulo p. 3.6. Zeros of Polynomials. 4. Field Extensions. 4.1. Field Extensions. 4.2. Rational Expressions. 4.3. Simple Extensions. 5. Simple Extensions. 5.1. Algebraic and Transcendental Extensions. 5.2. The Minimal Polynomial. 5.3. Simple Algebraic Extensions. 5.4. Classifying Simple Extensions. 6. The Degree of an Extension. 6.1. Definition of the Degree. 6.2. The Tower Law. 6.3. Primitive Element Theorem. 7. Ruler-and-Compass Constructions. 7.1. Approximate Constructions and More General Instruments. 7.2. Constructions in C. 7.3. Specific Constructions. 7.4. Impossibility Proofs. 7.5. Construction From a Given Set of Points. 8. The Idea Behind Galois Theory. 8.1. A First Look at Galois Theory. 8.2. Galois Groups According to Galois. 8.3. How to Use the Galois Group. 8.4. The Abstract Setting. 8.5. Polynomials and Extensions. 8.6. The Galois Correspondence. 8.7. Diet Galois. 8.8. Natural Irrationalities. 9. Normality and Separability. 9.1. Splitting Fields. 9.2. Normality. 9.3. Separability. 10. Counting Principles. 10.1. Linear Independence of Monomorphisms. 11. Field Automorphisms. 11.1. K-Monomorphisms. 11.2. Normal Closures. 12. The Galois Correspondence. 12.1. The Fundamental Theorem of Galois Theory. 13. Worked Examples. 13.1. Examples of Galois Groups. 13.2. Discussion. 14. Solubility and Simplicity. 14.1. Soluble Groups. 14.2. Simple Groups. 14.3. Cauchy

Über Autor(innen)

Ian Stewart is an emeritus professor of mathematics at the University of Warwick and a fellow of the Royal Society. Dr. Stewart has been a recipient of many honors, including the Royal Society’s Faraday Medal, the IMA Gold Medal, the AAAS Public Understanding of Science and Technology Award, and the LMS/IMA Zeeman Medal. He has published more than 210 scientific papers and numerous books, including several bestsellers co-authored with Terry Pratchett and Jack Cohen that combine fantasy with nonfiction.