A note of diagonalization of integral quadratic forms modulo p^m

Ali H. Hakami
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 17, 2011, Number 1, Pages 30–36
Full paper (PDF, 171 Kb)

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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 ℤ_p^m = ℤ / (p^m) be the ring of integers modulo p^m. Let Q(x) = Q(x₁, x₂, …, x_n) 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 p^m).

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.