Quadratic extensions of linearly compact fields
HTML articles powered by AMS MathViewer
- by Ron Brown and Hoyt D. Warner
- Trans. Amer. Math. Soc. 163 (1972), 379-399
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294307-6
- PDF | Request permission
Abstract:
A group valuation is constructed on the norm factor group of a quadratic extension of a linearly compact field, and the norm factor group is explicitly computed as a valued group. Generalizations and applications of this structure theory are made to cyclic extensions of prime degree, to square (and $p$th power) factor groups, to generalized quaternion algebras, and to quadratic extensions of arbitrary fields.References
- A. Adrian Albert, Structure of algebras, American Mathematical Society Colloquium Publications, Vol. XXIV, American Mathematical Society, Providence, R.I., 1961. Revised printing. MR 0123587
- N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 0194450 Ron Brown, Extended prime spots (in preparation). Ron Brown, D. K. Harrison and H. D. Warner, Ultracompletions at finite and infinite primes (in preparation).
- Paul F. Conrad, Embedding theorems for abelian groups with valuations, Amer. J. Math. 75 (1953), 1–29. MR 53933, DOI 10.2307/2372611
- Isidore Fleischer, Completeness in valued spaces and algebras, Quart. J. Math. Oxford Ser. (2) 15 (1964), 345–348. MR 177983, DOI 10.1093/qmath/15.1.345
- Ladnor Geissinger, A reciprocity law for maximal fields, Trans. Amer. Math. Soc. 125 (1966), 422–431. MR 204402, DOI 10.1090/S0002-9947-1966-0204402-2
- K. A. H. Gravett, Valued linear spaces, Quart. J. Math. Oxford Ser. (2) 6 (1955), 309–315. MR 87649, DOI 10.1093/qmath/6.1.309
- D. K. Harrison, Finite and infinite primes for rings and fields, Mem. Amer. Math. Soc. 68 (1966), 62. MR 207735 —, Seminar at the University of Oregon, 1967/68 (mimeographed notes).
- Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161 W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196.
- Solomon Lefschetz, Algebraic Topology, American Mathematical Society Colloquium Publications, Vol. 27, American Mathematical Society, New York, 1942. MR 0007093
- R. E. MacKenzie and G. Whaples, Artin-Schreier equations in characteristic zero, Amer. J. Math. 78 (1956), 473–485. MR 90584, DOI 10.2307/2372667
- Eben Matlis, Decomposable modules, Trans. Amer. Math. Soc. 125 (1966), 147–179. MR 201465, DOI 10.1090/S0002-9947-1966-0201465-5 O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der math. Wissenschaften, Band 117, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 27 #2485. A. Pfister, Quadratic forms, Notes by A. D. McGettrick, Cambridge University, Cambridge, 1967 (Xeroxed).
- Paulo Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Deuxième édition multigraphiée. MR 0249425
- O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776 E. Witt, Theorie der quadratischen formen in beliebigen korpern, J. Reine Angew. Math. 176 (1937), 31-44.
- Ron Brown, Valued vector spaces of countable dimension, Publ. Math. Debrecen 18 (1971), 149–151 (1972). MR 312202, DOI 10.5486/pmd.1971.18.1-4.20
- Ju. L. Eršov, On the elementary theory of maximal normed fields, Dokl. Akad. Nauk SSSR 165 (1965), 21–23 (Russian). MR 0190140
Bibliographic Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 163 (1972), 379-399
- MSC: Primary 12J20; Secondary 13A15
- DOI: https://doi.org/10.1090/S0002-9947-1972-0294307-6
- MathSciNet review: 0294307