Dettaglio del documento
Abstract algebra via numbers
Abstract
This book is a concise, self-contained treatise on abstract algebra with an introduction to number theory, where students normally encounter rigorous mathematics for the first time. The authors build up things slowly, by explaining the importance of proofs.
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Number theory with its focus on prime numbers is then bridged via complex numbers and linear algebra, to the standard concepts of a course in abstract algebra, namely groups, representations, rings, and modules. The interplay between these notions becomes evident in the various topics studied. Galois theory connects field extensions with automorphism groups. The group algebra ties group representations with modules over rings, also at the level of induced representations. Quadratic reciprocity occurs in the study of Fourier analysis over finite fields. Jordan decomposition of matrices is obtained by decomposition of modules over PID’s of complex polynomials. This latter example is just one of many stunning generalizations of the fundamental theorem of arithmetic, which in its various guises penetrates abstract algebra and figures multiple times in the extensive final chapter on modules. (Dal sito dell'editore)
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| eBook | ||
| Monografia | ||
| Description | *Abstract algebra via numbers / Lars Tuset. - Cham : Springer, 2025. - 1 testo elettronico (xix, 452 p.) | |
| ISBN E-Book | 9783031746239 | |
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| Subjects | 11-XX - Number theory [MSC 2020] | |
| 12-XX - Field theory and polynomials [MSC 2020] | ||
| 13-XX - Commutative algebra [MSC 2020] | ||
| 15-XX - Linear and multilinear algebra; matrix theory [MSC 2020] | ||
| 16-XX - Associative rings and algebras [MSC 2020] | ||
| Parole chiave |
Characters
Galois theory Group theory Linear algebra Modules Primes Representation Theory Thue's theorem |
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| Publication place | Cham | |
| Publisher | Springer <editore> | |
| Publication year | 2025 | |
| Thesaurus | 11-XX | |
| 12-XX | ||
| 13-XX | ||
| 15-XX | ||
| 16-XX | ||
| Associative rings and algebras [MSC 2020] | ||
| Commutative algebra [MSC 2020] | ||
| Field theory and polynomials [MSC 2020] | ||
| Linear and multilinear algebra; matrix theory [MSC 2020] | ||
| Number theory [MSC 2020] |