Fields and quadratic form schemes
Abstract
The paper presents a study of axiomatic theory of quadratic forms. Two operations on quadratic form schemes are investigated: the product of schemes and the group extension of schemes. The main result states that the product of schemes realized by fields is again realized by a field.
References
1. L. Brocker, Über die Anzahl der Anordnungen eines kommutativen Korpers, Arch. Math. 29 (1977), 458-464.
2. C.M. Cordes, Quadratic forms over non-real fields with a finite number of quaternion algebras, Pacific J. Math. 63 (1976), 357-366.
3. O. Endler, Valuation Theory, Springer-Verlag, New York 1972.
4. L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York—Paris 1963.
5. M. Kula, Fields with non-trivial Kaplansky’s radical and finite square class number, Acta Arith. 38 (1981), 411-418.
6. M. Kula, Fields with prescribed quadratic form schemes, Math. Z. 167 (1979), 201-212.
7. M. Kula, Ciala i schematy form kwadratowych (Fields and quadratic form schemes), Doctoral Thesis, Katowice 1979.
8. T.Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, Massachusetts 1973.
9. S. Lang, Algebra, Reading, Massachusetts, Addison-Wesley 1971.
10. M. Marshall, J. Yucas, Linked quaternionic mappings and their associated Witt rings, Pacific J. Math. 95 (1981), 411-425.
11. M. Marshall, Classification of finite spaces of orderings, Canad. J. Math. 31 (1979), 320-330.
12. A. Prestel, Remarks on the Pythagoreas and Hasse Number of Real Fields, J. Reine Angew. Math. 303/304 (1978), 284-294.
13. A. Sładek, Abstract Theory of Quadratic Forms, [w:] Prace matematyczne 10. Prace naukowe Uniwersytetu Śląskiego nr 332, pod red. K. Szymiczka, Katowice 1980, 46-57.
14. L. Szczepanik, Quaternion algebras and binary quadratic forms, [w:] Prace matematyczne 6. Prace naukowe Uniwersytetu Śląskiego nr 87, pod red. M. Kucharzewskiego, Katowice 1975, 17-27.
15. L. Szczepanik, Quadratic form schemes with non-trivial radical, (preprint).
16. L. Szczepanik, Fields and quadratic form schemes with the index of radical not exceeding 16, this issue.
2. C.M. Cordes, Quadratic forms over non-real fields with a finite number of quaternion algebras, Pacific J. Math. 63 (1976), 357-366.
3. O. Endler, Valuation Theory, Springer-Verlag, New York 1972.
4. L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York—Paris 1963.
5. M. Kula, Fields with non-trivial Kaplansky’s radical and finite square class number, Acta Arith. 38 (1981), 411-418.
6. M. Kula, Fields with prescribed quadratic form schemes, Math. Z. 167 (1979), 201-212.
7. M. Kula, Ciala i schematy form kwadratowych (Fields and quadratic form schemes), Doctoral Thesis, Katowice 1979.
8. T.Y. Lam, The algebraic theory of quadratic forms, Benjamin, Reading, Massachusetts 1973.
9. S. Lang, Algebra, Reading, Massachusetts, Addison-Wesley 1971.
10. M. Marshall, J. Yucas, Linked quaternionic mappings and their associated Witt rings, Pacific J. Math. 95 (1981), 411-425.
11. M. Marshall, Classification of finite spaces of orderings, Canad. J. Math. 31 (1979), 320-330.
12. A. Prestel, Remarks on the Pythagoreas and Hasse Number of Real Fields, J. Reine Angew. Math. 303/304 (1978), 284-294.
13. A. Sładek, Abstract Theory of Quadratic Forms, [w:] Prace matematyczne 10. Prace naukowe Uniwersytetu Śląskiego nr 332, pod red. K. Szymiczka, Katowice 1980, 46-57.
14. L. Szczepanik, Quaternion algebras and binary quadratic forms, [w:] Prace matematyczne 6. Prace naukowe Uniwersytetu Śląskiego nr 87, pod red. M. Kucharzewskiego, Katowice 1975, 17-27.
15. L. Szczepanik, Quadratic form schemes with non-trivial radical, (preprint).
16. L. Szczepanik, Fields and quadratic form schemes with the index of radical not exceeding 16, this issue.
KulaM. (1985). Fields and quadratic form schemes. Annales Mathematicae Silesianae, 1, 7-22. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14333
Mieczysław Kula
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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