S. K. PANDEY Abstract- In order to introduce some new examples (other than those usually found in textbooks) on groups, rings and...

#### S. K. PANDEY

Abstract- In order to introduce some new examples (other than those usually found in textbooks) on groups, rings and fields we usually make an attempt to create some examples. This helps to the students to learn the subject. If we just pose a question that write distinct examples of a finite field of order three. This creates curiosity and gives further insights into the subject. In the mathematical literature, the very common example of a finite field is Z

_{p.}#### GJSFR: Global Journals Blog

One does not find any other example in the textbooks. This leads us to write two simple articles entitled ‘A Note on Matrix Representations of Finite Cyclic Groups’ [1] and ‘Matrix Field of Finite and Infinite Order’ [2].
In the mean time an idea evolved that we can easily yield a finite field of order p

^{2}for every prime p≠2. This leads to write an article entitled ‘**Visualizing Finite Field of Order p**’ [3] which has been recently published in the^{2}through Matrices**Global Journals**. This article deals with a technique to produce a finite field p^{2}order without using the usual method of construction of finite fields.
The ideas to write above three articles evolved almost in a short period of time however it took years to realize the need to write and publish these works.

This article ([3]) may be found useful for those working on Cryptography. There are always computational problems associated with the finite fields of higher orders. However in the case of a matrix field of finite order the addition and multiplication in the field reduce to the usual addition and multiplication of matrices. Since [3] can be used to create a finite matrix field of order p

^{2}for each p>2 and therefore one can easily construct a finite field of as larger order as one needs to work.
In addition [3] is also useful to construct finite cyclic groups. Using the technique given in [3] one can yield a multiplicative cyclic group of order p

^{2}-1 for each p>2 .#### REFERENCES RÈFÈRENCES REFERENCIAS

[1] S. K. Pandey, A Note on Matrix Representations of Finite Cyclic Groups, International Journal of Mathematics and Its Applications, 4(1-B), 2016, 27-28.[2] S. K. Pandey, Matrix Field of Finite and Infinite Order, International Research Journal of Pure Algebra, 5(12), 2015, 214-216.

[3] S. K. Pandey, Visualizing Finite Field of Order p

^{2}through Matrices, Global Journal of Science Frontier Research (F), Volume XVI, Issue 1, Version1, 2016, 27-30.

Research article:

https://globaljournals.org/GJSFR_Volume16/3-Visualizing-Finite-Field.pdf