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Gaussian integer∗
Daume†
2013-03-21 12:24:42
A complex number of the form a + bi, where a, b ∈ Z, is called a Gaussian
integer.
It is easy to see that the set S of all Gaussian integers is a subring of C;
specifically, S is the smallest subring containing {1, i}, whence S = Z[i].
Z[i] is a Euclidean ring, hence a principal ring, hence a unique factorization
domain.
There are four units (i.e. invertible elements) in the ring Z[i], namely ±1
and ±i. Up to multiplication by units, the primes in Z[i] are
• ordinary prime numbers ≡ 3 mod 4
• elements of the form a ± bi where a2 + b2 is an ordinary prime ≡ 1 mod 4
(see Thue’s lemma)
• the element 1 + i.
Using the ring of Gaussian integers, it is not hard to show, for example, that
the Diophantine equation x2 + 1 = y 3 has no solutions (x, y) ∈ Z × Z except
(0, 1).
∗ hGaussianIntegeri created: h2013-03-21i by: hDaumei version: h30207i Privacy setting:
h1i hDefinitioni h11R04i h55-00i h55U05i h32M10i h32C11i h14-02i h18-00i
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