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In mathematics, an Eisenstein prime is an Eisenstein integer

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units {±1, ±ω, ±ω2}, a + bω itself and its associates.

The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.

The natural Eisenstein primes are exactly the natural primes end with 2, 5 or E (i.e. the natural primes congruent to 2 mod 3).

Factorization of numbers in Z[ω][]

The primes in Z[ω] with no imaginary part are exactly the primes in Z end with 2, 5 or E.

Since Z[ω] is a UFD (Unique Factorization Domain), thus all numbers in Z[ω] only have one factorization.

number factorization number factorization
1 unit 11 (3−ω) × (4+ω)
2 prime 12 2 × (−2+ω) × (−2+ω2)
3 (−1) × (1+2ω)2 13 (−1) × 5 × (1+2ω)2
4 22 14 24
5 prime 15 prime
6 (−1) × 2 × (1+2ω)2 16 2 × (1+2ω)4
7 (−2+ω) × (−2+ω2) 17 (5+2ω) × (5+2ω2)
8 23 18 22 × 5
9 (1+2ω)4 19 (−1) × (1+2ω)2 × (−2+ω) × (−2+ω2)
X 2 × 5 1X 2 × E
E prime 1E prime
10 (−1) × 22 × (1+2ω)2 20 (−1) × 23 × (1+2ω)2
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