A survey of trace forms of algebraic number fields:
Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore
World Scientific Pub. Co.
c1984
|
| Schriftenreihe: | Series in pure mathematics
v. 2 |
| Schlagworte: | |
| Online-Zugang: | FHN01 Volltext |
| Zusammenfassung: | Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester. These notes present the first systematic treatment of the trace form as an object in its own right. Chapter I discusses the trace form of F/Q up to Witt equivalence in the Witt ring W(Q). Special attention is paid to the Witt classes arising from normal extensions F/Q. Chapter II contains a detailed analysis of trace forms over p-adic fields. These local results are applied in Chapter III to prove that a Witt class X in W(Q) is represented by the trace form of an extension F/Q if and only if X has non-negative signature. Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form. It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone. Chapter V discusses the equivariant Witt theory of trace forms of normal extensions F/Q and Chapter VI relates the trace form of F/Q to questions of ramification in F. These notes were written in an effort to identify central problems. There are many open problems listed in the text. An introduction to Witt theory is included and illustrative examples are discussed throughout |
| Beschreibung: | ix, 316 p |
| ISBN: | 9789814412780 |
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| 490 | 0 | |a Series in pure mathematics |v v. 2 | |
| 520 | |a Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester. These notes present the first systematic treatment of the trace form as an object in its own right. Chapter I discusses the trace form of F/Q up to Witt equivalence in the Witt ring W(Q). Special attention is paid to the Witt classes arising from normal extensions F/Q. Chapter II contains a detailed analysis of trace forms over p-adic fields. These local results are applied in Chapter III to prove that a Witt class X in W(Q) is represented by the trace form of an extension F/Q if and only if X has non-negative signature. Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form. It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone. Chapter V discusses the equivariant Witt theory of trace forms of normal extensions F/Q and Chapter VI relates the trace form of F/Q to questions of ramification in F. These notes were written in an effort to identify central problems. There are many open problems listed in the text. An introduction to Witt theory is included and illustrative examples are discussed throughout | ||
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| author | Conner, P. E. 1932- |
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| author_variant | p e c pe pec |
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| dewey-ones | 512 - Algebra |
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| discipline | Mathematik |
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| isbn | 9789814412780 |
| language | English |
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| spelling | Conner, P. E. 1932- Verfasser aut A survey of trace forms of algebraic number fields P.E. Conner & R. Perlis Singapore World Scientific Pub. Co. c1984 ix, 316 p txt rdacontent c rdamedia cr rdacarrier Series in pure mathematics v. 2 Every finite separable field extension F/K carries a canonical inner product, given by trace(xy). This symmetric K-bilinear form is the trace form of F/K.When F is an algebraic number field and K is the field Q of rational numbers, the trace form goes back at least 100 years to Hermite and Sylvester. These notes present the first systematic treatment of the trace form as an object in its own right. Chapter I discusses the trace form of F/Q up to Witt equivalence in the Witt ring W(Q). Special attention is paid to the Witt classes arising from normal extensions F/Q. Chapter II contains a detailed analysis of trace forms over p-adic fields. These local results are applied in Chapter III to prove that a Witt class X in W(Q) is represented by the trace form of an extension F/Q if and only if X has non-negative signature. Chapter IV discusses integral trace forms, obtained by restricting the trace form of F/Q to the ring of algebraic integers in F. When F/Q is normal, the Galois group acts as a group of isometries of the integral trace form. It is proved that when F/Q is normal of prime degree, the integral form is determined up to equivariant integral equivalence by the discriminant of F alone. Chapter V discusses the equivariant Witt theory of trace forms of normal extensions F/Q and Chapter VI relates the trace form of F/Q to questions of ramification in F. These notes were written in an effort to identify central problems. There are many open problems listed in the text. An introduction to Witt theory is included and illustrative examples are discussed throughout Field extensions (Mathematics) Ring extensions (Algebra) Trace formulas Algebraischer Zahlkörper (DE-588)4068537-8 gnd rswk-swf Spurformel (DE-588)4182612-7 gnd rswk-swf Algebraischer Zahlkörper (DE-588)4068537-8 s Spurformel (DE-588)4182612-7 s 1\p DE-604 Perlis, R. Sonstige oth Erscheint auch als Druck-Ausgabe 9789971966041 Erscheint auch als Druck-Ausgabe 9971966042 Erscheint auch als Druck-Ausgabe 9971966050 http://www.worldscientific.com/worldscibooks/10.1142/0066#t=toc Verlag URL des Erstveroeffentlichers Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Conner, P. E. 1932- A survey of trace forms of algebraic number fields Field extensions (Mathematics) Ring extensions (Algebra) Trace formulas Algebraischer Zahlkörper (DE-588)4068537-8 gnd Spurformel (DE-588)4182612-7 gnd |
| subject_GND | (DE-588)4068537-8 (DE-588)4182612-7 |
| title | A survey of trace forms of algebraic number fields |
| title_auth | A survey of trace forms of algebraic number fields |
| title_exact_search | A survey of trace forms of algebraic number fields |
| title_full | A survey of trace forms of algebraic number fields P.E. Conner & R. Perlis |
| title_fullStr | A survey of trace forms of algebraic number fields P.E. Conner & R. Perlis |
| title_full_unstemmed | A survey of trace forms of algebraic number fields P.E. Conner & R. Perlis |
| title_short | A survey of trace forms of algebraic number fields |
| title_sort | a survey of trace forms of algebraic number fields |
| topic | Field extensions (Mathematics) Ring extensions (Algebra) Trace formulas Algebraischer Zahlkörper (DE-588)4068537-8 gnd Spurformel (DE-588)4182612-7 gnd |
| topic_facet | Field extensions (Mathematics) Ring extensions (Algebra) Trace formulas Algebraischer Zahlkörper Spurformel |
| url | http://www.worldscientific.com/worldscibooks/10.1142/0066#t=toc |
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