  The study of \textit{Dedekind Zeta Functions} over a number field extension
uses different aspects of both \textit{Algebraic} and \textit{Analytic Number
Theory}. In this paper, we shall learn about the structure and different
analytic aspects of such functions, namely the domain of its convregence and
analyticity at different points of $\mathbb{C}$ when the function is defined
over any finite field extension $K$ over $\mathbb{Q}$ . Moreover, given any two
Number Fields $L$ and $K$ over $\mathbb{Q}$ with $L$ being Normal over $K$, our
intention is to classify and study the primes in $K$ which split completely in
$L$. Also, we shall explore some special cases related to this result.
