CONGRUENT ZETA FUNCTIONS. NO.7

YOSHIFUMI TSUCHIMOTO

7.1. Jacobi symbol.

Definition 7.1. Let m be a positive odd integer. Let us factor m:

m = ∏_{i} p_{i}^{e_{i}}

where p_{i} are primes. Then for any n ∈ ℤ, we define Jacobi symbols as follows

(\frac{n}{m}) = ∏_{i} (\frac{n}{p_{i}} )^{e_{i}}

We further define

(\frac{a}{p}) = 0 if a ∈ pℤ.

Theorem 7.2 (quadratic reciprocity theorem). For any positive odd integers n, m, we have

(\frac{m}{n}) (\frac{n}{m}) = (−1)^{(m−1)(n−1)/4}.

Theorem 7.3. Let n be a postive odd integer. Then:

(1) (\frac{−1}{m}) = (−1)^{(m−1)/2}.

(2) (\frac{2}{m}) = (−1)^{(m^{2}−1)/8}.

Exercise 7.1. p = 113357 is a prime. (You may use the fact without proving it.) Is there any integer n such that

n^{2} = 11351 in ℤ/pℤ ?

If so, can you find such n?