  Cubic and biquadratic reciprocity have long since been referred to as "the
forgotten reciprocity laws", largely since they provide special conditions that
are widely considered to be unnecessary in the study of number theory. In this
exposition of finite fields and higher reciprocity, we will begin by
introducing concepts in abstract algebra and elementary number theory. This
will motivate our approach toward understanding the structure and then
existence of finite fields, especially with a focus on understanding the
multiplicative group $\mathbb{F}^{*}$. While surveying finite fields we will
provide another proof of quadratic reciprocity. We will proceed to investigate
properties of the general multiplicative character, covering the concept of a
general Gauss sum as well as basic notions of the Jacobi sum. From there we
will begin laying the foundations for the cubic reciprocity law, beginning with
a classification of the primes and units of the Eisenstein integers, denoted
$\mathbb{Z}[\omega]$, and further investigations into the residue class ring
$\mathbb{Z}[\omega]/\pi\mathbb{Z}[\omega]$ for $\pi$ prime, which is
predominantly the world in which cubic reciprocity lies. We then define the
cubic residue character and state the full law of cubic reciprocity. We will
finish the section on cubic reciprocity with a brief survey of the cubic
residue character of the even prime $2$ and state a significant result due to
Gauss that summarizes the conditions for $2$ to be a cubic residue. We conclude
with a brief survey of the law of biquadratic reciprocity.
