  In a simple graph $G$, we prove that the \textit{Hadwiger number}, $h(G)$, of
the given graph $G$ always upper bounds the \textit{chromatic number},
$\chi(G)$, of the given graph $G$, that is, $\chi(G) \leq h(G)$. This simply
stated problem is one of the fundamental questions in combinatorial
mathematics, which was made by Hugo Hadwiger in 1943. Consequently, it
independently verifies the most famous Four-Color Theorem: the case $h(G) = 4$
is equivalent to the Four-Color Theorem, that is, every planar graph is
$4$-colourable. In our novel approach, we use algebraic settings over a finite
field $\mathbb{Z}_p$. The algebraic setting, in essence, begins with the
complete graph with $h(G)$ vertices (which is a minor, $\mathcal{M}$, of the
given graph $G$) and iteratively extends to the simple graph $G$. This
conjecture has remained elusive, owing to a lack of understanding of the
interdependence, particularly the importance of Lemma 3.1, Lemma 3.2, Lemma
3.3, and Lemma 3.6 in Section 3.
