### On some computational and applications of finite fields

Jean Pierre Muhirwa

#### Abstract

Finite field is a wide topic in mathematics. Consequently, none can talk about the whole contents of finite fields. That is why this research focuses on small content of finite fields such as polynomials computational, ring of integers modulo p where p is prime or a power of prime. Most of the times, books which talk about finite fields are rarely to be found, therefore one can know how arithmetic computational on small finite fields works and be able to extend to the higher order. This means how integer and polynomial arithmetic operations are done for Z p such as addition, subtraction, division and multiplication in Z p followed by reduction of p (modulo p). Since addition is the same as subtraction and division is treated as the inverse of the multiplication, thus in this paper, only addition and multiplication arithmetic operations are applied for the considered small finite fields (Z 2 − Z 17 ). With polynomials, one can learn from this paper how arithmetic computational through polynomials over finite fields are performed since these polynomials have coefficients drawn from finite fields. The paper includes also construction of polynomials over finite fields as an extension of finite fields with polynomials. This lead to arithmetic computational tables for the finite fields F q [x]/f(x), where f(x) is irreducible over F q . From the past decades, many researchers complained about the applications of some topics in pure mathematics and therefore the finite fields play more important role in coding theory, which involves error-coding detection and error-correction as well as cyclic codes. As a result, this research paper shows these applications among others.

PDF

#### References

Cohen, A. M., Cuypers, H., & Sterk, H. (2013). Some tapas of computer algebra (Vol. 4). Springer Science & Business Media.

Cox, D. A. (2011). Galois theory. (Vol. 61). John Wiley & Sons.

Gong, G., Gyarmati, K., Hernando, F., Huczynska, S., Jungnickel, D., Kyureghyan, G. M., ... Shparlinski, I. E. (2013). Finite fields and their applications: character sums and polynomials (Vol. 11). Walter de Gruyter.

Karimian, Y., Ziapour, S., & Attari, M. A. (2012). Parity check matrix recognition from noisy codewords. arXiv preprint arXiv:1205.4641.

Lidl, R., & Niederreiter, H. (1994). Introduction to finite fields and their applications (2nd ed.). Cambridge University Press. doi: 10.1017/CBO9781139172769

Peterson, W. W., & Brown, D. T. (1961). Cyclic codes for error detection. Proceedings of the IRE, 49(1), 228–235.

Pless, V. (1998). Introduction to the theory of error-correcting codes (Vol. 48). John Wiley & Sons.

Roberts, J. A., & Vivaldi, F. (2005). Signature of time-reversal symmetry in polynomial automorphisms over finite fields. Nonlinearity, 18(5), 2171.

Rónyai, L. (1992). Galois groups and factoring polynomials over finite fields. SIAM Journal on Discrete Mathematics, 5(3), 345–365.

Sayed, M. (2011). Coset decomposition method for decoding linear codes. International Journal of Algebra, 5(28), 1395–1404.

Sziklai, P. (2013). Applications of polynomials over finite fields (Unpublished doctoral dissertation). ELTE TTK.

von zur Gathen, J., Shparlinski, I. E., & Stichtenoth, H. (n.d.). Finite fields: Theory and applications.

DOI: http://dx.doi.org/10.23755/rm.v39i0.521

### Refbacks

• There are currently no refbacks.

Copyright (c) 2020 Muhirwa Jean Pierre