A Note on Construction of Finite Field of Order $p$ and $p^2$

Abstract

In this note we construct finite field of order $p$ (here $p$ is a positive prime) and $p^{2}$ for $p>2$ through even square elements of $Z_{2p} $. It has been already noticed that a finite field of order $p^{2} $, $p>2$ can be directly constructed without using the concept of quotient rings. We utilize the same technique to yield a finite field of order $p^{2} $ for $p>2$ however here we use the notion of even square elements of a ring $R$. It is noticed that for all the finite fields constructed in this article the reducing modulo $m$ is a composite integer.