3 is not prime in Z[3]. Give it a little thought, and the result is not at all surprising. This is what 3 was invented for - 3 times 3 is 3. 5 is a prime number. But what about 2? Is it a prime? No, it is not. We can write

(1)	2 = (5 + 33)(-5 + 33),

where none of the factors is a unity (check their N-value.)

In fact, once we found one factorization, we can easily obtain many more. Recollect that U = 2 + 3 as well as U' = 2 - 3 are unities. Multiply the first factor in (1) by U and the second by U'. Apply the same procedure to the result:

(2)	2 = (5 + 33)(-5 + 33)   = (19 + 113)(-19 + 113)   = (71 + 413)(-71 + 413)...

Note that the N-value of any factor is 2. This implies that each of them is prime. So we learn not only that, in Z[3], 2 is not prime, but also that its factorization into prime factors is not unique. Indeed, in the presence of an infinite number of unities, no number may have a unique factorization. However, in (2) factors differ by a multiple of unity. Such primes are called associates. For example, 5 + 33 and 71 + 413 are associates, for (71 + 413) = (5 + 33)(7 + 43), where N(7 + 43) = 1 and 7 + 43 is a unity.

By the end of this page I plan to prove the following variant of the Fundamental Theorem of Arithmetic:

In Z[3], factorization of a number into a product of primes is unique up to the order of associated primes.

First let's establish an analogue of the Euclidean Algorithm:

Let A/B = x + y3, where x and y are rational. Any rational number p lies somewhere between [p] and [p]+1, where [p] is the largest integer smaller than p. Therefore, any rational number is at the distance of at most ½ from an integer. Let r and s be integers such that |x - r|½ and |y - s|½, respectively.

Let Q = r + s3 and R = B(x-r) + B(y-s)3. Then, of course, A = BQ + R, where A, B, Q all belong to Z[3] and so does R = A - BQ. Let's verify that |N(R)| < |N(B)|.

N(R) = N(B)((x-r)² - 3(y-s)²). By the int_domain3.shtml,

(4)	|(x-r)2 - 3(y-s)2||x-r|2 + 3|y-s|2(½)2 + 3(½)2 = 1.

Equality is only achieved when both |x-r| and |y-s| equal ½. But in this case,

|(x-r)2 - 3(y-s)2| = |½2 - 3½2| = ½ < 1

Therefore always |(x-r)² - 3(y-s)²| < 1, and we have |N(R)| < |N(B)|.

(3) is a very powerful result. Now we may start an analogue of Euclid's Algorithm with confidence, assured that it will terminate in a finite number of steps. When it stops, we shall have found a number D with the property that

1. D|A and D|B, and
2. For any C such that C|A and C|B, we also have C|D.

This D is naturally called the greatest common divisor of A and B, and is denoted as D = gcd(A, B). As in N, the algorithm may be extended to yield two numbers S and T such that

The greatest common divisor is obviously defined up to a factor of unity. Two numbers are called coprime (mutually prime) iff their greatest common divisor is a unity. Since 1 associates with every unity,

for coprime A and B and some S and T; exactly as was the case with integers.

We are in a position to prove the most fundamental property of division:

Uniqueness of the prime factorization follows immediately from (5). Let A = P₁P₂...P_n = Q₁Q₂...Q_m, where all P's and Q's are prime. P₁ divides Q₁Q₂...Q_m. Therefore, by (5), it divides one of the Q's. Both being prime, they are associates of each other and their ratio is a unity. Assume P₁|Q₁ and Q₁/P₁ = U. Replace Q₂ with UQ₂. This will not affect the primality of Q₂. We can chip off factors from one side or another until we get, for example, Q₁Q₂...Q_k = 1, all other factors from the two sides having been matched. The remaining Q's are all unities, hence, not prime. Whence k = 0, n = m, and the proof is completed.

Remark

6 = 3·2 = 3(3-1) = 3² - 3 = (3 + 3)(3 - 3). Since 2|6, we also have that 2|(3 + 3)(3 - 3). However, 2(3 + 3) and 2(3 - 3), for, say, (3 + 3)/2 = 3/2 + 3/2 which does not belong to Z[3] at all. Combined with (4), we get another proof that 2 is not prime in Z[3].

Question

Does or does not the identity 3·2 = (3 + 3)(3 - 3) contradict the fact that the prime factorization in Z[3] is unique?

Reference

1. H.M.Stark, An Introduction to Number Theory, The MIT Press, 1995, Ninth printing.

• Strange Integers
  • Divisors and Primes
  • Fundamental Theorem of Arithmetic
  • Gaussian Integers
  • Remarks and Examples
• Constructible Numbers
  • Examples
  • Four Problems of Antiquity
  • A Property of Cubic Equations