Divisor Theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Boston, MA
Birkhäuser Boston
1990
|
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Man sollte weniger danach streben, die Grenzen der mathematischen Wissenschaften zu erweitern, als vielmehr danach, den bereits vorhandenen Stoff aus umfassenderen Gesichtspunkten zu betrachten - E. Study Today most mathematicians who know about Kronecker's theory of divisors know about it from having read Hermann Weyl's lectures on algebraic number theory [We], and regard it, as Weyl did, as an alternative to Dedekind's theory of ideals. Weyl's axiomatization of what he calls "Kronecker's" theory is built- as Dedekind's theory was built- around unique factorization. However, in presenting the theory in this way, Weyl overlooks one of Kronecker's most valuable ideas, namely, the idea that the objective of the theory is to define greatest common divisors, not to achieve factorization into primes. The reason Kronecker gave greatest common divisors the primary role is simple: they are independent of the ambient field while factorization into primes is not. The very notion of primality depends on the field under consideration- a prime in one field may factor in a larger field- so if the theory is founded on factorization into primes, extension of the field entails a completely new theory. Greatest common divisors, on the other hand, can be defined in a manner that does not change at all when the field is extended (see §1.16). Only after he has laid the foundation of the theory of divisors does Kronecker consider factorization of divisors into divisors prime in some specified field |
| Beschreibung: | 1 Online-Ressource (XIV, 166 p) |
| ISBN: | 9780817649777 9780817649760 |
| DOI: | 10.1007/978-0-8176-4977-7 |
Internformat
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Datensatz im Suchindex
| _version_ | 1804153089482031104 |
|---|---|
| any_adam_object | |
| author | Edwards, Harold M. |
| author_facet | Edwards, Harold M. |
| author_role | aut |
| author_sort | Edwards, Harold M. |
| author_variant | h m e hm hme |
| building | Verbundindex |
| bvnumber | BV042419170 |
| classification_tum | MAT 000 |
| collection | ZDB-2-SMA ZDB-2-BAE |
| ctrlnum | (OCoLC)860044339 (DE-599)BVBBV042419170 |
| dewey-full | 512.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.6 |
| dewey-search | 512.6 |
| dewey-sort | 3512.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-0-8176-4977-7 |
| format | Electronic eBook |
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| id | DE-604.BV042419170 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:04Z |
| institution | BVB |
| isbn | 9780817649777 9780817649760 |
| language | English |
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| spelling | Edwards, Harold M. Verfasser aut Divisor Theory by Harold M. Edwards Boston, MA Birkhäuser Boston 1990 1 Online-Ressource (XIV, 166 p) txt rdacontent c rdamedia cr rdacarrier Man sollte weniger danach streben, die Grenzen der mathematischen Wissenschaften zu erweitern, als vielmehr danach, den bereits vorhandenen Stoff aus umfassenderen Gesichtspunkten zu betrachten - E. Study Today most mathematicians who know about Kronecker's theory of divisors know about it from having read Hermann Weyl's lectures on algebraic number theory [We], and regard it, as Weyl did, as an alternative to Dedekind's theory of ideals. Weyl's axiomatization of what he calls "Kronecker's" theory is built- as Dedekind's theory was built- around unique factorization. However, in presenting the theory in this way, Weyl overlooks one of Kronecker's most valuable ideas, namely, the idea that the objective of the theory is to define greatest common divisors, not to achieve factorization into primes. The reason Kronecker gave greatest common divisors the primary role is simple: they are independent of the ambient field while factorization into primes is not. The very notion of primality depends on the field under consideration- a prime in one field may factor in a larger field- so if the theory is founded on factorization into primes, extension of the field entails a completely new theory. Greatest common divisors, on the other hand, can be defined in a manner that does not change at all when the field is extended (see §1.16). Only after he has laid the foundation of the theory of divisors does Kronecker consider factorization of divisors into divisors prime in some specified field Mathematics Algebra Category Theory, Homological Algebra Mathematik Algebraische Kurve (DE-588)4001165-3 gnd rswk-swf Schiefkörper (DE-588)4052359-7 gnd rswk-swf Algebraische Zahlentheorie (DE-588)4001170-7 gnd rswk-swf Divisor (DE-588)4150324-7 gnd rswk-swf Teilbarkeit (DE-588)4203392-5 gnd rswk-swf Arithmetik (DE-588)4002919-0 gnd rswk-swf Divisor (DE-588)4150324-7 s Algebraische Zahlentheorie (DE-588)4001170-7 s 1\p DE-604 Algebraische Kurve (DE-588)4001165-3 s 2\p DE-604 Arithmetik (DE-588)4002919-0 s 3\p DE-604 Teilbarkeit (DE-588)4203392-5 s 4\p DE-604 Schiefkörper (DE-588)4052359-7 s 5\p DE-604 https://doi.org/10.1007/978-0-8176-4977-7 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Edwards, Harold M. Divisor Theory Mathematics Algebra Category Theory, Homological Algebra Mathematik Algebraische Kurve (DE-588)4001165-3 gnd Schiefkörper (DE-588)4052359-7 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Divisor (DE-588)4150324-7 gnd Teilbarkeit (DE-588)4203392-5 gnd Arithmetik (DE-588)4002919-0 gnd |
| subject_GND | (DE-588)4001165-3 (DE-588)4052359-7 (DE-588)4001170-7 (DE-588)4150324-7 (DE-588)4203392-5 (DE-588)4002919-0 |
| title | Divisor Theory |
| title_auth | Divisor Theory |
| title_exact_search | Divisor Theory |
| title_full | Divisor Theory by Harold M. Edwards |
| title_fullStr | Divisor Theory by Harold M. Edwards |
| title_full_unstemmed | Divisor Theory by Harold M. Edwards |
| title_short | Divisor Theory |
| title_sort | divisor theory |
| topic | Mathematics Algebra Category Theory, Homological Algebra Mathematik Algebraische Kurve (DE-588)4001165-3 gnd Schiefkörper (DE-588)4052359-7 gnd Algebraische Zahlentheorie (DE-588)4001170-7 gnd Divisor (DE-588)4150324-7 gnd Teilbarkeit (DE-588)4203392-5 gnd Arithmetik (DE-588)4002919-0 gnd |
| topic_facet | Mathematics Algebra Category Theory, Homological Algebra Mathematik Algebraische Kurve Schiefkörper Algebraische Zahlentheorie Divisor Teilbarkeit Arithmetik |
| url | https://doi.org/10.1007/978-0-8176-4977-7 |
| work_keys_str_mv | AT edwardsharoldm divisortheory |