LEADER 03005nam 22005895 450 001 9910983306603321 005 20250703131539.0 010 $a9783031746239 010 $a3031746236 024 7 $a10.1007/978-3-031-74623-9 035 $a(CKB)36812840000041 035 $a(MiAaPQ)EBC31812115 035 $a(Au-PeEL)EBL31812115 035 $a(DE-He213)978-3-031-74623-9 035 $a(OCoLC)1475039086 035 $a(EXLCZ)9936812840000041 100 $a20241202d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aAbstract Algebra via Numbers /$fby Lars Tuset 205 $a1st ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (462 pages) 311 08$a9783031746222 311 08$a3031746228 327 $aChapter 1. Number theory -- Chapter 2. Construction of numbers -- Chapter 3. Linear algebra -- Chapter 4. Groups -- Chapter 5. Representations of finite groups -- Chapter 6. Rings -- Chapter 7. Field extensions -- Chapter 8. Galois theory -- Chapter 9. Modules -- Chapter 10. Appendix. 330 $aThis 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. 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. 606 $aAlgebra 606 $aNumber theory 606 $aAlgebra 606 $aNumber Theory 606 $aÀlgebra$2thub 606 $aTeoria de nombres$2thub 608 $aLlibres electrònics$2thub 615 0$aAlgebra. 615 0$aNumber theory. 615 14$aAlgebra. 615 24$aNumber Theory. 615 7$aÀlgebra 615 7$aTeoria de nombres 676 $a512 700 $aTuset$b Lars$01252224 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910983306603321 996 $aAbstract Algebra via Numbers$94316247 997 $aUNINA