How to have 4 subsections on 2 paragraphs

The words "subsections" and "paragraphs" are here quite misleading because in TeX they refer to text elements, while you are writing equations. This might be a possible realization (I'm not sure where \chi_0 is supposed to be)

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
\chi_0(x) &= 1 \\[1ex]
\chi_1(x) &=
\begin{cases}
    1,   & \text{if } x = 0, \\
    a,   & \text{if } x = 1, \\
    a^2, & \text{if } x = 2, \\
    a^3, & \text{if } x = 3, \\
    a^4, & \text{if } x = 4, 
\end{cases}
&
\chi_3(x) &=
\begin{cases}
    1,   & \text{if } x = 0, \\
    a^3, & \text{if } x = 1, \\
    a,   & \text{if } x = 2, \\
    a^4, & \text{if } x = 3, \\
    a^2, & \text{if } x = 4, 
\end{cases}
\\[1ex]
\chi_2(x) &=
\begin{cases}
    1,   & \text{if } x = 0, \\
    a^2, & \text{if } x = 1, \\
    a^4, & \text{if } x = 2, \\
    a,   & \text{if } x = 3, \\
    a^3, & \text{if } x = 4, 
\end{cases}
&
\chi_4(x) &=
\begin{cases}
    1,   & \text{if } x = 0, \\
    a^4, & \text{if } x = 1, \\
    a^3, & \text{if } x = 2, \\
    a^2, & \text{if } x = 3, \\
    a,   & \text{if } x = 4.
\end{cases}
\end{align*}

\end{document}

As campa stated, first showed question without code was misleading, and in this case can be nicely solved with align* (+1).

So this answer is not to search the acceptance, but just to show what you can with the commented minipages in other cases, not only with maths, since the minipages can contain anything, even a true subsection (e.g. \subsection{text} Text ...) with images, tables, etc. But of course, also is possible only with your equations:

\documentclass[12pt]{article}
\usepackage{amsmath}
\begin{document}

Let us define the following:


\[\chi_0(x) = 1\]

\noindent
\begin{minipage}[t]{.45\textwidth}
$\chi_1(x)$ = $\begin{cases}
    1,   & \text{if } x = 0, \\
    a,   & \text{if } x = 1, \\
    a^2, & \text{if } x = 2, \\
    a^3, & \text{if } x = 3, \\
    a^4, & \text{if } x = 4, 
\end{cases}$\bigskip
\end{minipage}\hfill%
\begin{minipage}[t]{.45\textwidth}
$\chi_3(x)$ = $\begin{cases}
    1,   & \text{if } x = 0, \\
    a^3, & \text{if } x = 1, \\
    a,   & \text{if } x = 2, \\
    a^4, & \text{if } x = 3, \\
    a^2, & \text{if } x = 4, 
\end{cases}$\bigskip
\end{minipage}\hfill%
\begin{minipage}[t]{.45\textwidth}
$\chi_2(x)$ = $\begin{cases}
    1,   & \text{if } x = 0, \\
    a^2, & \text{if } x = 1, \\
    a^4, & \text{if } x = 2, \\
    a,   & \text{if } x = 3, \\
    a^3, & \text{if } x = 4, 
\end{cases}$\bigskip
\end{minipage}\hfill%
\begin{minipage}[t]{.45\textwidth}
$\chi_4(x)$ = $\begin{cases}
    1,   & \text{if } x = 0, \\
    a^4, & \text{if } x = 1, \\
    a^3, & \text{if } x = 2, \\
    a^2, & \text{if } x = 3, \\
    a,   & \text{if } x = 4, 
\end{cases}$\bigskip
\end{minipage}\hfill%

\end{document}