Gaussian integers Z[i] = {a+bi : a,b∈Z}. Norm N(a+bi) = a²+b². Units: ±1, ±i. Gaussian prime: irreducible in Z[i]. A rational prime p: splits if p≡1(mod 4), stays prime if p≡3(mod 4), ramifies if p=2=(1+i)(1−i)(−i).
How to Use
Enter two Gaussian integers:
- a+bi: First number
- c+di: Second number
- Operations: +, ×, norm, GCD
Gaussian Primes
π∈Z[i] is prime if N(π) is prime, OR π = p (rational prime with p≡3 mod 4). Example: 2+i is prime (N=5), 3 is prime in Z[i] (stays prime), but 5=(2+i)(2−i) splits.
Applications
- Sums of two squares
- Fermat's theorem on primes
- Signal processing (QAM)
- Algebraic number theory
Step-by-Step Instructions
- 1Enter a+bi.
- 2Enter c+di.
- 3Choose operation.
- 4View result.
- 5Check Gaussian primality.