# What is the intuition behind Gauss sums?

Let $\chi$ be a character on the field $F_p$, and fix some $a \in F_p$. We define a Gauss sum to be: $g_a (\chi) = \sum_{t\in F_p}\chi(t)\zeta^{at}$ where $\zeta$ is a primitive $p^{th}$ root of unity.

What is the intuition behind this definition?

When you say $$\chi$$ is a character of the field $$F_p$$, you really mean it is a character of the group $$F_p^\times$$. Any (multiplicative) character $$\chi$$ on $$F_p^\times$$ can be extended to a function on $$F_p$$ by setting $$\chi(0) = 0$$, and with this convention $$\chi$$ as a function on $$F_p$$ is totally multiplicative. Fixing a choice of nontrivial $$p$$th root of unity $$\zeta$$, any function $$f \colon F_p \rightarrow {\mathbf C}$$ has a Fourier transform $${\mathcal F}f \colon F \rightarrow {\mathbf C}$$ given by $$({\mathcal F}f)(a) = \sum_{t \in F_p} f(t)\overline{\zeta^{at}}$$. So the Gauss sum $$g_a(\chi)$$ is essentially the Fourier transform at $$a$$ of the function $$\chi$$ (as a function on $$F$$). For more on Fourier transforms of functions on a finite abelian group, see Section 4 (starting at Definition 4.4) of
Another intuition (besides the idea that a Gauss sum of a character is basically the Fourier transform of that character, viewed as a function on the additive group $$F_p$$) is that a Gauss sum is a discrete analogue of the Gamma function. See pp. 56--58 of Koblitz's book "$$p$$-adic Analysis: A Short Course on Recent Work" for a table illustrating this analogy (including the idea that a Jacobi sum is like the Beta function).
One explanation is given in this math.SE answer. In the language of that answer, you want to describe the unique quadratic subfield of $\mathbb{Q}(\zeta_p)$. Since it's quadratic, it's generated by the square root of some rational, so the Galois group acts by multiplication by $-1$ on it. Therefore you want to find an element of $\mathbb{Q}(\zeta_p)$ such that the Galois group $(\mathbb{Z}/p\mathbb{Z})^{\ast}$ acts by multiplication by $-1$ on it, and up to a constant that element is a Gauss sum (for the quadratic character). Gauss sums with respect to more general characters have a similar relationship to actions of the Galois group.
They are the discrete Fourier transform of the character $\chi$ (which presumably is a multiplicative character -- like the quadratic symbol, no?). This allows one to express $\chi$ as a sum of exponentials.