Gaussian integer

A Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z. This is a Euclidean domain which cannot be turned into an ordered ring.

Related Topics:
Complex number - Integer - Integral domain - Euclidean domain - Ordered ring

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Formally, Gaussian integers are the set

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:{a+bi | a,bin mathbb{Z} }.

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The norm of a Gaussian integer is the natural number defined as

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:N(a + bi) = a2 + b2.

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The norm is multiplicative, i.e.

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:N(z·w) = N(z)·N(w).

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The units of Z are therefore precisely those elements with norm 1, i.e. the elements

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:1, −1, i and −i.

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The prime elements of Z are also known as Gaussian primes. Some prime numbers (which, by contrast, are sometimes referred to as "rational primes") are not Gaussian primes; for example 2 = (1 + i)(1 − i) and 5 = (2 + i)(2 − i).

Related Topics:
Prime element - Prime number

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Those rational primes which are congruent to 3 (mod 4) are Gaussian primes; those which are congruent to 1 (mod 4) are not. This is because primes of the form 4k + 1 can always be written as the sum of two squares (Fermat's theorem), so we have

Related Topics:
Mod - Fermat's theorem

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:p = a2 + b2 = (a + bi)(a − bi).

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If the norm of a Gaussian integer z is a prime number, then z must be a Gaussian prime, since every non-trivial factorization of z would yield a non-trivial factorization of the norm. So for example 2 + 3i is a Gaussian prime since its norm is 4 + 9 = 13. This implies that since there are infinitely many ordinary primes then there must be infinitely many Gaussian primes.

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The ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q(i) consisting of the complex numbers whose real and imaginary part are both rational.

Related Topics:
Integral closure - Field - Gaussian rationals - Rational

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