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Quadratic forms over fields

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Rozprawy Matematyczne tom/nr w serii: 152 wydano: 1977
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Abstrakty
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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
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Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 152
Liczba stron
63
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CLII
Daty
wydano
1977
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autor
Bibliografia
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  • [2] C. M. Cordes, The Witt group and the equivalence of fields with respect to quadratic forms, J. Algebra 26 (1973), pp. 400-421.
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  • [4] R. Elman and T. Y. Lam, Pfister forms and K-theory of fields, J. Algebra 23 (1972), pp. 181-213.
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  • [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.
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  • [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.
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  • [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.
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  • [24] W. Scharlau, Quadratic Forms, Queen's Papers on Pure and Applied Mathematics No. 22, Kingston, Ont., 1969.
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  • [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.
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  • [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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