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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 or
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.
The Gaussian integers are the set
In other words, a Gaussian integer is a complex number such that its real and imaginary parts are both integers.
Since the Gaussian integers are closed under addition and multiplication, they form a commutative ring, which is a subring of the field of complex numbers. It is thus an integral domain.
When considered within the complex plane, the Gaussian integers constitute the 2-dimensional integer lattice.
The conjugate of a Gaussian integer a + bi is the Gaussian integer a – bi.
The norm of a Gaussian integer is its product with its conjugate.
The norm of a Gaussian integer is thus the square of its absolute value as a complex number. The norm of a Gaussian integer is a nonnegative integer, which is a sum of two squares. Thus a norm cannot be of the form 4k + 3, with k integer.
The norm is multiplicative, that is, one has
for every pair of Gaussian integers z, w. This can be shown directly, or by using the multiplicative property of the modulus of complex numbers.
The units of the ring of Gaussian integers (that is the Gaussian integers whose multiplicative inverse is also a Gaussian integer) are precisely the Gaussian integers with norm 1, that is, 1, –1, i and –i.
Gaussian integers have a Euclidean division (division with remainder) similar to that of integers and polynomials.
Official source
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