# 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?

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.
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 this, see exercise 7 on page 16 at http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/charthy.pdf
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).
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.