Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1