imported>Praemonitus: /* Basic definitions */ +ref.

2025-10-26T17:17:26Z

Basic definitions: +ref.

imported>ThighFish: /* Gaussian primes */

2025-05-05T07:01:52Z

Gaussian primes

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← Previous revision		Revision as of 17:17, 26 October 2025
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	Gaussian integers are named after the German mathematician [[Carl Friedrich Gauss]].		Gaussian integers are named after the German mathematician [[Carl Friedrich Gauss]].

	[[File:Gaussian integer lattice.svg\|thumb\|217px\|Gaussian integers as [[lattice point]]s in the [[complex plane]]]]		[[File:Gaussian integer lattice.svg\|thumb\|217px\|Gaussian integers as [[integer lattice point]]s in the [[complex plane]]]]

	==Basic definitions==		==Basic definitions==
	The Gaussian integers are the set<ref name="Fraleigh 1976 286"/>		The Gaussian integers are the set<ref name="Fraleigh 1976 286"/><ref>{{cite journal \| title=Exploring the Gaussian Integers \| first=Robert G. \| last=Stein \| journal=The Two-Year College Mathematics Journal \| volume=7 \| issue=4 \| pages=4–10 \| doi=10.1080/00494925.1976.11974454 }}</ref>
	:<math>\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.</math>		:<math>\mathbf{Z}[i]=\{a+bi \mid a,b\in \mathbf{Z} \}, \qquad \text{ where } i^2 = -1.</math>

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	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]].		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 {{math\|2}}-dimensional [[~~integer~~ lattice]].		When considered within the [[complex plane]], the Gaussian integers constitute the {{math\|2}}-dimensional [[square lattice]].

	The ''conjugate'' of a Gaussian integer {{math\|''a'' + ''bi''}} is the Gaussian integer {{math\|''a'' − ''bi''}}.		The ''conjugate'' of a Gaussian integer {{math\|''a'' + ''bi''}} is the Gaussian integer {{math\|''a'' − ''bi''}}.
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	If the norm of {{math\|''g''}} is even, then either {{math\|1=''g'' = 2<sup>''k''</sup>''h''}} or {{math\|1=''g'' = 2<sup>''k''</sup>''h''(1 + ''i'')}}, where {{math\|''k''}} is a positive integer, and {{math\|''N''(''h'')}} is odd. Thus, one chooses the associate of {{math\|''g''}} for getting a {{math\|''h''}} which fits the choice of the associates for elements of odd norm.		If the norm of {{math\|''g''}} is even, then either {{math\|1=''g'' = 2<sup>''k''</sup>''h''}} or {{math\|1=''g'' = 2<sup>''k''</sup>''h''(1 + ''i'')}}, where {{math\|''k''}} is a positive integer, and {{math\|''N''(''h'')}} is odd. Thus, one chooses the associate of {{math\|''g''}} for getting a {{math\|''h''}} which fits the choice of the associates for elements of odd norm.

	==Gaussian primes==		==<span class="anchor" id="Gaussian primes"></span>Gaussian primes==
	As the Gaussian integers form a [[principal ideal domain]], they also form a [[unique factorization domain]]. This implies that a Gaussian integer is [[irreducible element\|irreducible]] (that is, it is not the product of two [[unit (ring theory)\|non-unit]]s) if and only if it is [[prime element\|prime]] (that is, it generates a [[prime ideal]]).		As the Gaussian integers form a [[principal ideal domain]], they also form a [[unique factorization domain]]. This implies that a Gaussian integer is [[irreducible element\|irreducible]] (that is, it is not the product of two [[unit (ring theory)\|non-unit]]s) if and only if it is [[prime element\|prime]] (that is, it generates a [[prime ideal]]).

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	This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated. Therefore, a third method is often preferred for hand-written computations. It consists in remarking that the norm {{math\|''N''(''d'')}} of the greatest common divisor of {{math\|''a''}} and {{math\|''b''}} is a common divisor of {{math\|''N''(''a'')}}, {{math\|''N''(''b'')}}, and {{math\|''N''(''a'' + ''b'')}}. When the greatest common divisor {{math\|''D''}} of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing {{math\|''D''}}.		This method of computation works always, but is not as simple as for integers because Euclidean division is more complicated. Therefore, a third method is often preferred for hand-written computations. It consists in remarking that the norm {{math\|''N''(''d'')}} of the greatest common divisor of {{math\|''a''}} and {{math\|''b''}} is a common divisor of {{math\|''N''(''a'')}}, {{math\|''N''(''b'')}}, and {{math\|''N''(''a'' + ''b'')}}. When the greatest common divisor {{math\|''D''}} of these three integers has few factors, then it is easy to test, for common divisor, all Gaussian integers with a norm dividing {{math\|''D''}}.

	For example, if {{math\|1=''a'' = 5 + 3''i''}}, and {{math\|1=''b'' = 2 − 8''i''}}, one has {{math\|1=''N''(''a'') = 34}}, {{math\|1=''N''(''b'') = 68}}, and {{math\|1=''N''(''a'' + ''b'') = 74}}. As the greatest common divisor of the three norms is 2, the greatest common divisor of {{math\|''a''}} and {{math\|''b''}} has 1 or 2 as a norm. As a gaussian integer of norm 2 is ~~necessary~~ associated to {{math\|1 + ''i''}}, and as {{math\|1 + ''i''}} divides {{math\|''a''}} and {{math\|''b''}}, then the greatest common divisor is {{math\|1 + ''i''}}.		For example, if {{math\|1=''a'' = 5 + 3''i''}}, and {{math\|1=''b'' = 2 − 8''i''}}, one has {{math\|1=''N''(''a'') = 34}}, {{math\|1=''N''(''b'') = 68}}, and {{math\|1=''N''(''a'' + ''b'') = 74}}. As the greatest common divisor of the three norms is 2, the greatest common divisor of {{math\|''a''}} and {{math\|''b''}} has 1 or 2 as a norm. As a gaussian integer of norm 2 is necessarily associated to {{math\|1 + ''i''}}, and as {{math\|1 + ''i''}} divides {{math\|''a''}} and {{math\|''b''}}, then the greatest common divisor is {{math\|1 + ''i''}}.

	If {{math\|''b''}} is replaced by its conjugate {{math\|1=''b'' = 2 + 8''i''}}, then the greatest common divisor of the three norms is 34, the norm of {{math\|''a''}}, thus one may guess that the greatest common divisor is {{math\|''a''}}, that is, that {{math\|''a'' {{!}} ''b''}}. In fact, one has {{math\|1=2 + 8''i'' = (5 + 3''i'')(1 + ''i'')}}.		If {{math\|''b''}} is replaced by its conjugate {{math\|1=''b'' = 2 + 8''i''}}, then the greatest common divisor of the three norms is 34, the norm of {{math\|''a''}}, thus one may guess that the greatest common divisor is {{math\|''a''}}, that is, that {{math\|''a'' {{!}} ''b''}}. In fact, one has {{math\|1=2 + 8''i'' = (5 + 3''i'')(1 + ''i'')}}.
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	Most of the unsolved problems are related to distribution of Gaussian primes in the plane.		Most of the unsolved problems are related to distribution of Gaussian primes in the plane.

	*[[Gauss's circle problem]] does not deal with the Gaussian integers per se, but instead asks for the number of [[lattice point]]s inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.		*[[Gauss's circle problem]] does not deal with the Gaussian integers per se, but instead asks for the number of [[integer lattice point]]s inside a circle of a given radius centered at the origin. This is equivalent to determining the number of Gaussian integers with norm less than a given value.

	There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:		There are also conjectures and unsolved problems about the Gaussian primes. Two of them are:

Gaussian integer - Revision history

imported>Praemonitus: /* Basic definitions */ +ref.

imported>ThighFish: /* Gaussian primes */