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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## Details

### 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* _{pm}* = ℤ / (

*p*) be the ring of integers modulo

^{m}*p*. Let

^{m}*Q*(

**x**) =

*Q*(

*x*

_{1},

*x*

_{2}, …,

*x*) be a nonsingular quadratic form with integer coefficients. In this paper we shall prove that any nonsingular quadratic form

_{n}*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

- Larry J. Gerstein,
*Basic Quadratic Forms*, American Mathematical Society, 2008. - G. L. Watson,
*Integral Quadratic Forms*, Cambridge University Press, 1960. - Michel Artin,
*Algebra*, Prentice-Hall, New Jersey, 1991. - 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

APAHakami, A. H. (2011). A note of diagonalization of integral quadratic forms modulo *p ^{m}*, Notes on Number Theory and Discrete Mathematics, 17(1), 30-36.

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

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