Kronecker symbol (a/n): extends Jacobi/Legendre to all integers. For odd positive n: same as Jacobi. For n=2: (a/2)=0 if a even, 1 if a≡±1(mod 8), −1 if a≡±3(mod 8). For n=−1: (a/−1)=−1 if a<0, 1 if a≥0. For n=0: (a/0)=1 if |a|=1, else 0.
How to Use
Enter a and n:
- (a/n): 0, 1, or −1
- Decomposition: Product formula
- Table: For range of values
Extensions
Legendre (a/p) for odd prime p → Jacobi (a/n) for odd positive n → Kronecker (a/n) for ALL integers n. Each step relaxes restrictions on the bottom. Kronecker handles n=0, negative n, even n.
Quadratic Forms
For discriminant D of a quadratic form, (D/n) determines representation: can n = ax²+bxy+cy² with discriminant D? The Kronecker symbol is the natural character for quadratic forms over Z.
Step-by-Step Instructions
- 1Enter a.
- 2Enter n.
- 3Compute (a/n).
- 4See decomposition.
- 5View table.