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Concept
Gaussian integer
Summary
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf{Z}[i] or \Z[i].
Gaussian integers share many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization and many related properties. However, Gaussian integers do not have a total ordering that respects arithmetic.
Gaussian integers are algebraic integers and form the simplest ring of quadratic integers.
Gaussian integers are named after the German mathematician Carl Friedrich Gauss.
Basic definitions
The Gaussian integers are the set
:\mathbf{Z}[i]={a+bi \mid a,b\in \mathbf{Z} }, \qquad \text{ where } i^2 = -1.
In other words, a Gaussian integer is a complex numbe
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