Number Theory
November 13, 2006
Today’s dose of Gaussian Integers…
1. Ordinary integers are also Gaussian integers (by the identification a a + 0i).
We call them rational integers, and if they are prime (in the ordinary integers) we call
them rational primes. Of course it’s being real that distinguishes them, not being
rational; but these are standard terms.
As we have seen, rational primes may or may not be Gaussian primes. (3 and 7 are, 2
and 5 aren’t.)
2. What are the Gaussian primes?
a. 1 + i is a prime, and so are its associates –1+i, 1 – i, and –1 – i.
a. Any rational prime p that satisfies p 3 (mod 4) is a Gaussian prime, and so
are its associates –p, pi, and –pi. (Proof: N(p) = p2, and any nontrivial factor would
have to have norm p, but no Gaussian integer has norm p because a2 + b2 = p never
occurs.)
b. If a2 + b2 = p, where now p is a rational prime satisfying p 1 (mod 4), then
a + bi is a Gaussian prime. Primes of this form come in eights: There are a + bi and its
associates, and the complex conjugate a – bi and its associates. (Proof: Since
N(a + bi) = p, there is no room for a nontrivial factor.)
Are there any other Gaussian primes? No, but I don’t think it’s obvious. The proof
would be easier after we prove unique factorization, but I propose to skip the proof then,
too.
3. Unique factorization in the Gaussian primes.
We needed ordering to prove unique factorization in Z. Specifically, we needed it for
these steps:
(a) If a | b, and b is positive, then a ≤ b.
(b) Given a and b, if b isn’t 0, we can write a = q b + r, with 0 ≤ r < b.
(In Z, q and r were unique, but we didn’t need that uniqueness to prove unique
factorization.)
We don’t have ordering in the Gaussian integers, but we have the norm, which turns out
to be just as good.
(a) If (a + bi) | (c + di) then N(a + bi) ≤ N(c + di).
(b) Given Gaussian integers x and y, if y isn’t 0, we can write x = q y + r, with
N(r) ≤ N(y).
(For the proof of the last part, see text p. 554. It uses geometry and isn’t hard. The
numbers q and r are not always unique.)
With these adjustments, the proof of unique factorization in the Gaussian integers works
exactly as it does for Z:
(a) Given non-zero Gaussian integers x and y, we can use the Euclidean
algorithm to find their gcd.
- Actually the gcd isn’t unique; if d is a gcd of x and y, then so are –d,
di, and –di. But it’s unique up to associates. We could have had the same problem in Z.
For example, the Euclidean algorithm could have given us +7 or –7 for gcd(14,21), but
we just chose the positive value.
- The gcd is “greatest” in the strong sense: Any common divisor of x and
y is a divisor of their gcd. (Or, of any of their gcds!)
(b) If gcd(x, y) = 1, we can reverse the algorithm to write ax + by = 1 for some
Gaussian integers a and b.
(c) Now if p is a Gaussian prime and p | xy, then p | x or p | y. Proof: If p
doesn’t divide x, then gcd (p, x) = 1, and we can write ap + bx = 1 for some a and b, or
apy + bxy = y. Now p divides both terms on the left, so it divides y.
(d) Any Gaussian integer can be factored into primes. If it isn’t prime then it can
be factored, and if the factors aren’t prime then they can be factored in turn, and the
process can’t go on forever because the factors’ norms keep getting smaller.
(e) If x = p1p2…pk = q1q2…qh, then each one of the pi’s must divide one of the
qj’s, and so on. (It isn’t necessary that pi actually be one of the qj’s; it could be one of the
associates of one of the qj’s. So in the Gaussian integers, factorization is unique only up
to associates. That’s true in the integers, too: 21 = 3 7 = ( -3 ) ( -7 ).)
4. An Application
Theorem: In the ordinary integers, if p is a prime and p 1 (mod 4), then p can be
written in as a sum of two squares, p = a2 + b2, in exactly one way (except for
exchanging a and b, and reversing the signs of a and/or b).
Proof. We showed earlier that p can be written as a sum of two primes. Suppose that
p = a2 + b2 and also p = c2 + d2. Then a bi and c di are all primes in the Gaussian
integers (because they all have norm p). Now
p = (a + bi) (a – bi) and p = (c + di) (c – di)
are two different prime factorizations of p in the Gaussian integers, so they must really
be the same, except for the order of the factors and substituting associates of the factors.//
So understanding the Gaussian integers helps us understand ordinary integers. This
keeps happening.
5. Restatement of the Gaussian primes.
Given the theorem, we can restate the list of Gaussian primes:
a. (1 + i) and its associates.
b. Each rational prime that is congruent to 3 (mod 4).
c. For each rational prime that is congruent to 1 (mod 4), exactly two Gaussian
primes: a + bi and a – bi, and their associates, where a and b are chosen so that
a2 + b2 = p.
(end)