A note of diagonalization of integral quadratic forms modulo pm

Ali H. Hakami
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 1, Pages 30—36
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Authors and affiliations

Ali H. Hakami
Department of Mathematics, Ahmadu Bello University
Zaria, Nigeria

Abstract

Let m be a positive integer, p be an odd prime, and ℤpm = ℤ / (pm) be the ring of integers modulo pm. Let Q(x) = Q(x1, x2, …, xn) be a nonsingular quadratic form with integer coefficients. In this paper we shall prove that any nonsingular quadratic form Q(x) over ℤ, Q(x) is equivalent to a diagonal quadratic form (modulo pm).

Keywords

  • Integral quadratic form
  • Nonsingular quadratic form
  • Diagonalization quadratic form modulo prime

AMS Classification

  • 11E08

References

  1. Larry J. Gerstein, Basic Quadratic Forms, American Mathematical Society, 2008.
  2. G. L. Watson, Integral Quadratic Forms, Cambridge University Press, 1960.
  3. Michel Artin, Algebra, Prentice-Hall, New Jersey, 1991.
  4. R. Lidl and H. Niederreiter, Encyclopedia of Mathematics and its Applications, Finite Fields, Addison-Wesley Publishing Company, 1983.

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Cite this paper

APA

Hakami, A. H. (2011). A note of diagonalization of integral quadratic forms modulo pm, Notes on Number Theory and Discrete Mathematics, 17(1), 30-36.

Chicago

Hakami, Ali H. “A Note of Diagonalization of Integral Quadratic Forms Modulo pm.” Notes on Number Theory and Discrete Mathematics 17, no. 1 (2011): 30-36.

MLA

Hakami, Ali H. “A Note of Diagonalization of Integral Quadratic Forms Modulo pm.” Notes on Number Theory and Discrete Mathematics 17.1 (2011): 30-36. Print.

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