I want to know properties of the following sum: $$\sum\_{j=0}^{p-1} \omega^{\beta j^2}= ~? $$
where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and $p-1$). If anyone has any references related to this, let me know. 


Also, if someone can tell me about the sum:
$$\sum\_{j=0}^{p-1} \omega^{\beta j^n}= ~? $$
for higher values of $n$ such as 3, that would also be nice.



Summation formulas involving roots of unity to various powers
Your first sum is a special Gauss sum. For its value in general, see Corollary 9.16 in Montgomery-Vaughan: Multiplicative number theory I. Your second sum can also be expressed in terms of Gauss sums (associated with primitive characters modulo $p$), and in particular its absolute value is at most $(n-1)\sqrt{p}$, assuming $p\nmid\beta$. See Exercise 4c in Section 9.2.1 of the same book. 


