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Zawartość
Pełne teksty:
Warianty tytułu
Abstrakty
CONTENTS
Introduction.................................................................................................................................. 5
Chapter I. Preliminaries oil round and Pfister forms.......................................................... 8
Chapter II. Basic properties of the Grothendieck ring......................................................... 11
§ 1. Prime ideals in G(F)...................................................................................................... 11
§ 2. Elements of special types............................................................................................ 15
§ 3. Local properties of G(F)................................................................................................ 20
Chapter III. Group structure of G(F) and W(F)....................................................................... 23
§1. General decomposition theorems.............................................................................. 23
§ 2. Value sets of binary forms and the group structure of G(F) and W(F)................. 28
§ 3. The rank of G(F) and W(F)............................................................................................ 32
Chapter IV. Equivalence of fields with respect to quadratic forms.................................... 36
§ 1. G-equivalences.............................................................................................................. 36
§ 2. W-equivalences.............................................................................................................. 41
§ 3. Comparisons.................................................................................................................. 43
Chapter V. A Galois correspondence in the quadratic form theory.................................. 46
§ 1. The binary case............................................................................................................. 46
§ 2. A generalization.............................................................................................................. 54
Chapter VI. Field constructions................................................................................................ 57
Chapter VII. Open problems..................................................................................................... 60
References.................................................................................................................................. 62
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
152
Liczba stron
63
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLII
Daty
wydano
1977
Twórcy
autor
Bibliografia
- [1] J. K. Arason und A. Pfister, Beweis des Krullschen Durchschnittsalzes für den Wittring, Invent. Math. 12 (1971), pp. 173-176.
- [2] C. M. Cordes, The Witt group and the equivalence of fields with respect to quadratic forms, J. Algebra 26 (1973), pp. 400-421.
- [3] R. Elman, Round forms over real algebraic function fields in one variable, Proc. Amer. Math. Soc. 41 (1973), pp. 431-436.
- [4] R. Elman and T. Y. Lam, Pfister forms and K-theory of fields, J. Algebra 23 (1972), pp. 181-213.
- [5] R. Elman and T. Y. Lam, Quadratic forms over formally real fields and pythagorean fields, Amer. J. Math. 94 (1972), pp. 1155-1194.
- [6] R. Elman and T. Y. Lam, Quadratic forms and the u-invariant, II, Invent. Math. 21 (1973), pp. 125-137.
- [7] R. Elman, T. Y. Lam and A. Prestel, On some Hasse principles over formally real fields, Math. Zeitschr. 134 (1973), pp. 291-301.
- [8] L. Fuchs, Infinite Abelian Groups, vol. I, Academic Press, New York and London 1970.
- [9] D. K. Harrison, Witt Rings, University of Kentucky Notes, Lexington 1970.
- [10] J. S. Hsia and R. P. Johnson, Round and group quadratic forms over global fields, J. Number Theory 5 (1973), pp. 356-366.
- [11] I. Kaplansky, Fröhlich's local quadratic forms, J. Reine Angew. Math. 239/240 (1969), pp. 74-77.
- [12] M. Knebusch, A. Rosenberg, and R. Ware, Structure of Witt rings and quotients of abelian group rings, Amer. J. Math. 94 (1972), pp. 119-155.
- [13] M. Kula, Ordered fields and quadratic forms over pythagorean fields, Uniw. Śląski w Katowicach Prace Naukowe — Prace Mat. 7 (1977), pp. 13-21.
- [14] A. G. Kurosh, Lectures on General Algebra, Chelsea, New York 1963.
- [15] T. Y. Lam, The Algebraic Theory of Quadratic Forms, Benjamin, Reading 1973.
- [16] S. Lang, Algebra, 3rd ed., Addison-Wesley, Reading 1971.
- [17] F. Lorenz, Quadratische Formen über Körpern, Lecture Notes in Mathematics No. 130, Springer, Berlin 1970.
- [18] F. Lorenz und J. Leicht, Die Primideale des Wittschen Ringes, Invent. Math. 10 (1970), pp. 82-88.
- [19] M. Marshall, Round quadratic forms, Math. Zeitschr. 140 (1974), pp. 255-262.
- [20] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970), pp. 318-344.
- [21] J. Milnor and D. Hussmoller, Symmetric Bilinear Forms, Springer, Berlin-Heidelberg-New York 1973.
- [22] O. T. O'Meara, Introduction to Quadratic Forms, 2nd ed., Springer, Berlin-Heidelberg-New York 1971.
- [23] A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1 (1966), pp. 116-132.
- [24] W. Scharlau, Quadratic Forms, Queen's Papers on Pure and Applied Mathematics No. 22, Kingston, Ont., 1969.
- [25] A. Sładek, Grothendieck groups of quadratic forms over formally real fields, Uniw. Śląski w Katowicach Prace Naukowe — Prace Mat. 5 (1974), pp. 41-47.
- [26] T. A. Springer, Quadratic forms over fields with a discrete valuation, I, Indag. Math. 17 (1955), pp. 352-362.
- [27] K. Szymiczek, Grothendieck groups of quadratic forms and 6-equivalence of fields, Proc. Cambridge Philos. Soc. 73 (1973), pp. 29-36. Corrigendum, ibid. 74 (1973), p. 199.
- [28] K. Szymiczek, Universal binary quadratic forms, Uniw. Śląski w Katowicach Prace Naukowe — Prace Mat. 5 (1974), pp. 49-57.
- [29] K. Szymiczek, Quadratic forms over fields with finite square class number, Acta Arith. 28 (1975), pp. 195-221.
- [30] K. Szymiczek, A structure theorem for Grothendieck group of quadratic forms over a field, Colloquia Mathematica Societatis János Bolyai, Vol. 13, Topics in Number Theory, Debrecen (Hungary), 1974, North-Holland Publ, Co., Amsterdam-Oxford-New York 1976, pp. 389-397.
- [31] E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), pp. 31-44.
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