# Expressing a sum involving a nontrivial character as a Jacobi sum

Let $\chi$ be a multiplicative non trivial character of $F_p$ and $\rho$ be a character of order 2. Show that $\sum\chi(1-t^2)=J(\chi,\rho).$
[Hint: Evaluate $J(\chi,\rho)$ using the relation $N(x^2=a)=1+\rho(a)$]

I have read Rosen on my own, and I have understood characters and Jacobi sums, but can't do this question. Please give me some hints.

Note that $\rho$ is not just any character of order 2, but the unique character of order $2$, i.e. $\rho(a) = \left( \frac{a}{p}\right)$. Then, as in the hint, we have $N(x^2=a) = 1 + \rho(a)$, because $x^2=a$ either has 0 solutions, or 2 distinct solutions modulo $p$, and so in either case exactly $1 + \left(\frac{a}{p}\right)$ solutions.

So we have $$J(\chi,\rho)=\sum_{a \not\equiv 0,1} \chi(1-a) \rho(a) = \sum_a \chi(1-a) (N(x^2=a)-1) = \sum_a \chi(1-a)N(x^2=a) - \sum_a \chi(1-a).$$

Now since $\chi$ is a non-trivial character, its sum over ${1,2,...,p-1}$ is 1, therefore its sum over $\{2,...,p-1\}$, i.e. the second term above, is equal to $0$.

So now $$J(\chi,\rho)=\sum_{a \not\equiv 0,1} \chi(1-a)N(x^2=a) = \sum_{a \in S} 2\chi(1-a),$$

where $S$ is the set of square residues mod $p$ except 0 or 1, i.e. $S = \{ b^2: 1< b< p\}$. Noting that each square residue $b^2\in S$ occurs twice as $b$ ranges over $\{1<b<p\}$ we can write

$$\sum_{a\in S} 2\chi(1-a) = \sum_{1 < b < p} \chi(1-b^2).$$

• thanks a lot. i have one more thing to ask as u seem good in number theory. i have done burton for elementary number theory and the course i have this semester has contents mainly following Rosen. But no other text like zuckerman etc use gauss sums and jocobi sums and characters. So if i instead study nivan & zuckerman, will it be ok. coz then i wont be able to prove questions like this. I found zuckerman more intresting as i started reading it and read first two chapters. what do you suggest. Commented Sep 8, 2014 at 2:30
• Both textbooks are excellent, but it depends how far you want to pursue number theory. Niven-Zuckerman-Montgomery covers elementary number theory very well, and then tiptoes into more advanced topics just to give a taste. Ireland-Rosen goes much further in every topic, since it's an introduction to Modern number theory after all. The books are written for different, if overlapping, audiences. NZM is in my opinion the best introduction to elementary number theory among several great choices, and IR is the best if you want to pursue it further. Commented Sep 8, 2014 at 3:01
• Elementary number theory really is the tip of a giant magnificent iceberg. The transition from elementary number theory to algebraic or analytic number theory can be a little bit rough, because one is suddenly faced with a multitude of abstract ideas to absorb (and care about.) As far as I know, IR is the only book that tries to make this transition smooth by showing how the modern ideas develop naturally from the classical ones. The fact that it's very well-written makes it extra special. Commented Sep 8, 2014 at 3:02
• On the other hand, it's probably best only to read things you actually find interesting. IR is so very thorough that perhaps one needs to skip some topics and return to them after seeing more advanced stuff in order to appreciate them. Commented Sep 8, 2014 at 3:02
• For example, I didn't really care for Gauss sums very much until I tried to prove for myself the simplest case of the Kronecker-Weber theorem from algebraic number theory. This is a theorem that is the starting point for much of modern number theory and current research, and in its simplest case it says the square root of every integer can be written as a sum of roots of unity with rational coefficients. Gauss sums do exactly this. Commented Sep 8, 2014 at 3:03