On the group of form classes of large degree
https://doi.org/10.22405/2226-8383-2024-25-5-113-125
Abstract
Groups of form classes were introduced in Number Theory by Gauss, for binary quadratic forms. He defined the notions of equivalence and composition and introduced a group structure in classes of equivalence for the family of quadratic forms with discriminants not divisible by a square of integral number. Further investigations of Gauss were developed in various directions. One of them is a generalization of the theory to multivariate quadratic forms, in which widely studied questions on representation of integral numbers by various quadratic forms. Other direction concerns the notion of composition. But with the growth of the number of variables the question stands very difficult. In 1898, A. Hurwits showed that for quadratic forms with the number of variables greater than 8, it is hard to introduce suitable notion of composition.
This result of A. Hurwits was explaned by Y. V. Linnik from non-associative algebras’ point of a view. It is established that the notion of discriminant for forms of high degree is not so substantive as for quadratic forms. Sometimes, strict difference between forms having one and the same discriminant, is well known. To overcome these difficulties, it is convenient to consider forms connected with given extension of the field.
References
1. Edwards, H. M. 1977, “The Last Theorem of Fermat: A General Introduction to Algebraic
2. Number Theory”, Springer, New York.
3. Postnikov, M. M. 1982, “Introduction to the Theory of Algebraic Numbers”, Nauka, Moscow.
4. Mal’tsev, I. A. “Linear Algebra: 2nd ed.”, Lan, St. Petersburg. 384 p.
5. Weil, A. 1940, “Algebraic Number Theory”, Princeton University Press. 225 p.
6. Borevich, Z. I. and Shafarevich, I. R. 1972, “Number Theory: 2nd ed.”, Nauka, Moscow.
7. Gurwitz, A. 1898, “On the composition of quadratic forms of large numbers of variables”, News of G¨ottingen, pp. 309–316.
8. Linnik, Yu. V. 1938, “Generalization of the Frobenius theorem and establishing a connection
9. with Hurwitz’s theorem on the composition of quadratic forms”, Izvestiya of the Academy of
10. Sciences of the USSR, Series Mathematics, 2, pp. 41–52.
11. Alburt, A. 1942, “Quadratic forms allowing composition”, Annals of Mathematics, 43(1), pp.
12. –177.
13. Dubish, R. 1946, “Composition of quadratic forms”, Annals of Mathematics, 47(3), pp. 510–527.
14. Kostrikhin, A. I. 2009, “Introduction to Algebra”, Nauka, Moscow. 495 p.
15. Lang, S. 1968, “Algebra”, Mir, Moscow. 564 p.
16. Gelfand, I. M. 1971, “Lectures on Linear Algebra”, Nauka, Moscow. 271 p.
17. Mal’tsev, A. I. 2005, “Foundations of Linear Algebra”, Nauka, Moscow. 470 p.
18. Bourbaki, N. 2000, “Commutative Algebra”, Mir, Moscow.
19. Rodossky, K. A. 1988, “Euclid’s Algorithm”, Nauka, Moscow.
Review
For citations:
Meshaik S.A. On the group of form classes of large degree. Chebyshevskii Sbornik. 2024;25(5):113-125. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-5-113-125
Views:
314
Email this article 