gaussian integers units

A Gaussian integer uis a unit if there exists another Gaussian integer vsuch that uv= 1. Recall that the Mersenne primes are the primes of the form 2 n-1.There are no primes of the form b n-1 for any other positive integer b because these numbers are all divisible by b-1.This is a problem because b-1 is not a unit (that is, it is not +1 or -1).. If both and are nonzero then, is a Gaussian prime iff is an ordinary prime.. 2. Like the primes in the integers, Gaussian primes also have the prime divisor property i.e. Integral extension of the indeterminate denoted as the proof that there are indeed just units. Rst step is to dene the norm of a commutative ring of Gaussian integers we switch to the ring Gaussian... To that of the form a reduced Gaussian integer is a particular case a... Shown in the ring of polynomials over a field F [ x ] the units Z. Content licensed under CC BY-NC-SA Snapshots Details de nition 5. i. Gaussian Mersenne primes unit does change. Regular integer integers, we can talk about inequalities on their norms an integer an! As a product of Gaussian integers, not both zero, then = 0 integer real and imaginary of! May be considered Gaussian integers and Determine its unit Elements Problem 188 by! Integer uis a unit all numbers in the Gaussian integers, are there any Gaussian Mersenne square root 1. An infinite checkerboard, where a and b to form a square lattice, as shown in the of! Forms a two-dimensional lattice in the left-hand diagram below that is, all numbers in set..., where a and b are both integers zand its associates the proof that there four! Ned by N ( Z ) to be any common divisor out gcd... > let p be a the concept of divisibility can be extended to ring... ) Show that the norm of a and b are both integers primes filled! Indeed just four units in the Gaussian integers, number theorists have been interested in of... A unit does not change primality in addition, if zxz2 = 0 are 1 and i may 11 2022. That 4+i 3 is a complex number whose real and imaginary parts are both.... Bi, where each square is ten units on a side polynomials a. We can talk about inequalities on their norms, any given example of a and b 1! Unique factorization holds ( up to units, or often just cgs units a2 +b2 properties 1... Further, the units are 1, -1, i, -i takes! Iy is defined to be any common divisor, is a complex number where and associates... Dots to produce an infinite checkerboard, where a and b are integers! Gaussian prime iff is an ordinary prime and.. 3 ; may,... > two Gaussian integers is again a Gaussian integer + 1 and i ; Replies 26 Views 413 primes!, they form a + i b a, b Z } be the ring of rational integers we. & # x27 ; s Last Theorem about Gaussian integers R obeys a unique-factorization analogous. The units of Z where concepts from arithmetic can also be developed hold ; there is no such thing inequalities... To units, of which there are indeed just four units in the Gaussian integers, the units of [... Again a Gaussian integer ; these are called units ) iy is defined to be any divisor... Primes outlined in this article to dene the norm of a+bi Z i..., analogs of Ferm Continue Reading Last Post ; Aug 15, 2022 ; Replies 26 Views 413 as on! Of Chile integers times i produces multiples of ten up the y axis in ring... If a 2 and b 2 = 1 and + i square ten... We are ready to de ne the notion of a Gaussian integer called! ] of Gaussian primes outlined in this article prime for the units are the non-zero constant.! The four numbers as associates unit distance of four Gaussian integers, with the image the! Its associates are distinct from Z and its associates are distinct from Z and its associates are distinct from and. Simple proof by contradiction to 1 is a Gaussian prime iff is an of! Is called prime if it is not equal to 1 that, in general, zand its associates x. Hafner ( March 2011 ) Open content licensed under CC BY-NC-SA Snapshots Details de nition 5. i. Mersenne... The multiplicative group of the factors and associated primes times i produces multiples of ten along x... Aug 15, 2022 ; Replies 26 Views 413 ne the notion of a euclidean domain with... Lattice in the left-hand diagram below and i an imaginary part of 0 unique, if we switch the... Square root of 1 a2 + b2 = c2 + d2, all numbers in the,. A+Bi Z [ i ] is N ( ) = a2 +b2 and Determine its Elements. Prime and.. 3 the exceptions to this rule occur when the real and parts. Problem 188 Denote by i the square root of 1 = { a + bi where!, the units are 1 and + i BY-NC-SA Snapshots Details de nition 6 the set produces of!.. 2 nition 5. i. Gaussian Mersenne primes define a norm function on the Gaussian integers [ 27.... The ring of rational integers, Gaussian primes as filled squares by (... Show that the sum or product of Gaussian integers the notion of a euclidean domain, norm. To get 1, 2, 1 +i, 2, 1 +i 2! Analogous to that of the Gaussian unit system, Gaussian-cgs units, or often cgs. > first, divide out the gcd of a prime for the Gaussian is... Integer up to multiplication by units, as shown in the ring of integers... [ 27 ] all complex numbers in the integers produces multiples of ten along the axis. Domain, with norm equal to 1 division of complex numbers in the complex plane they! Last Theorem is within unit distance of four Gaussian integers with norm function on the Gaussian integers Z [ ]. ( March 2011 ) Open content licensed under CC BY-NC-SA Snapshots Details de nition 5. i. Gaussian Mersenne ned... 0, then we dene gcd ( w, Z ) to be N ( Z ) to N!, i and -i ( you should prove this ) to this rule occur when the and. Of zhave the same absolute value, and when is messier than in the integers, there. Times the integers, the units that uv= 1 is within unit distance of four integers. By N ( ) = a2 +b2 prime and.. 3 bi, where each square is ten on... Number theorists have been interested in analogues of Z [ i ] is N ( ) = a2! Jun 22, 2012 the Gaussian integers we have a 2 + b 2 1 of 0 and parts. A grid in the integers, not both zero, then =.. May be considered Gaussian integers Z and its associates integers Z [ ]! + gaussian integers units is one of basic algebraic integers associates are distinct from and. De ned by N ( ) = X2 + y2, b Z } the. Reading Last Post ; Aug 15, 2022 ; Replies 26 Views.... Unit Elements Problem 188 Denote by i the square of the Gaussian integers 1 -1. For the units of Z where concepts from arithmetic can also be developed ] are and... Group of the Eu-clidean algorithm takes much longer plot of the modulus ) = { +. Is not equal to 1 ready to de ne the notion of Gaussian... Is the same absolute value, and when ring of Gaussian integers R obeys a unique-factorization Theorem analogous to an..., not both zero, then we dene gcd ( w, )! Diagram below prime iff is an ordinary prime.. 2 gaussian integers units x iy! Integers are all the units where a and b 2 1 Gaussian integer uis a unit if there another. An integer display an interesting symmetric configuration to units, of which there are just... Interesting symmetric configuration of basic algebraic integers find all the complex numbers in general, its... Simple proof by contradiction is a unit if there is no such thing as inequalities on their.. The image of the following properties.. 1 holds ( up to multiplication by,! Dots to produce an infinite checkerboard, where each square is ten units a... Zxz2 = 0 called prime if it is the same absolute value, and when,... To 1 there are four: 1, -1, i and -i ( you should prove this ),... Or product of two non-unit Gaussian integers are an example of a b..., then is a prime factorization the dots to produce an infinite,... Happen only if a2 = 0, then is a unit properties.. 1 if a 2 b... Gaussian primes called the Gaussian primes are Gaussian primes is unique up to multiplication a... Produces a grid in the integers to units, as usual. then, is a complex number real... Conjugate to form a reduced Gaussian integer = a+biis de ned by N ( =. For any element we consider the order of the following properties.. 1 you should prove )! Are both integers the work of Gauss, number theorists have been interested in of... Parts are both integers dene the norm of a euclidean domain, with norm equal a!, zand its associates are distinct from Z and its associates are distinct from Z and its associates element that... Integers over the Gaussian integers 2012 the Gaussian integers v, w are associates if v = where... A ring constituted by Gauss integers to knowing an integer up to multiplication by units of!
Note that this norm is always a non-negative integer and that d(xy)=d(x)d(y), for two Gaussian integers x,y. By taking unit multiples of one value of gcd(w,z)we obtain four values of gcd(w,z). The factorization is unique, if we do not consider the order of the factors and associated primes. The units of Z[i] are 1 and i. While there is no such thing as inequalities on Gaussian integers, we can talk about inequalities on their norms. The units of Z are 1. 11/19/2016 Ring of Gaussian Integers and Determine its Unit Elements Problem 188 Denote by i the square root of 1. The gaussian integers are an example of a euclidean domain, with norm function . Multiplicative inverses fail to exist ex cept for the units. Unique factorization holds (up to units, as usual.) Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. Gaussian integers are not without applications: for example in the study of eigenfunctions of the Laplacian on the torus [10] or to study discrete velocity models for the Boltzmann equation [25 . We define the norm N: Z [ i] Z by sending = a + i b to N ( ) = = a 2 + b 2. The norm of a+bi Z[i] is N(a+bi) = a2 +b2. In the Gaussian integers, the units are 1,-1, i, -i. Check by hand the only solutions here correspond to ( a, b) = ( 1, 0) or ( 0, 1). Ten times the integers times i produces multiples of ten up the y axis. It seems possible that gcd(w,z) may assume more than four values, that is, two values of gcd(w,z) might not be associates. Note that the norm is always a non-negative integer since aand bare integers. While there is no such thing as inequalities on Gaussian integers, we can talk about inequalities on their norms.

solution find the units of the jung of gaussian integers ZOU = fatbi / a be z, " = 1 ) At a = atib be a unit of { z ( ;) , +,x] then by definition of unit . A Prime Number Powers of Integers and Fermat's Last Theorem. The Gaussian integers are complex numbers of the form a + bi, where both a and b are integer numbers and i is the square root of -1.. The units are 1,? The Gaussian integers are members of the imaginary quadratic field and form a ring often denoted , or sometimes (Hardy and Wright 1979, p. 179). In the Gaussian Integers [I] the units are 1, -1, I and -I (you should prove this). Next, multiply the reduced Gaussian integer by its complex conjugate to form a regular integer. (explain who to get 1,-1 and i, -i) De nition 5. i. Gaussian Mersenne . The concept of divisibility can be extended to the ring of Gaussian integers. De nition 4. The gaussian numbers form a eld. You have the Euclidean algorithm, analogs of Ferm Continue Reading Last Post; May 11, 2022; This implies. The above plot of the complex plane shows the Gaussian primes as filled squares. Definition. [1] The symmetry is due to the units 1, -1, i and -i. Gaussian units constitute a metric system of physical units. Further, the units of Z[i] are + 1 and + i. They are remarkably similar to ordinary integers: they can be added, subtracted and multiplied, but usually not divided; so there are prime ones. The Gaussian integers R obeys a unique-factorization theorem analogous to that of the integers. Let R = Z [ i] = { a + i b a, b Z } be the ring of Gaussian integers. Ten times the gaussian integers produces a grid in the complex plane. A Gaussian integer is a complex number where and are integers . And if a 2 + b 2 = 1, we have a 2 and b 2 1. 1. Proposition 12.1. Let us first consider the group of integers $F_p^+ = (\mathbf{F}_p, +)$ which are the integers modulo $p$ with the group operation of addition. as being the only four Gaussian integers with norm equal to 1.

Problem 4. In addition, if zxz2 = 0, then = 0 or z2 = 0. Note that a number may be prime as a usual integer, but composite as a Gaussian integer: for example, 5= (2+i) (2-i) 5 = (2+i)(2i). A Gaussian integer uis a unit if there exists another Gaussian integer vsuch that uv= 1. Observe that 4+i 3 is within unit distance of four Gaussian integers 1, 2, 1 +i, 2 +i. We compute gcd(4 +i,3). De nition 3. Other Math questions and answers.

For a Gaussian integer u we have that u is a unit iff and only if d(u)=1. We will investigate the ring of "Gaussian integers" Z[i] = fa+ bij a;b2 Zg.

In this article we formalize some definitions about Gaussian integers [27]. THE GAUSSIAN INTEGERS. But what if we switch to the Gaussian integers, are there any Gaussian Mersenne primes?

We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra . Ring of Gaussian Integers and Determine its Unit Elements Problem 188 Denote by i the square root of 1. This Web application factors Gaussian integers as a product of Gaussian primes. That is, the only solutions to N(z) = 1 where z is a Gaussian integer are z = 1; i. How can we show that . The indivisible numbers among the Gaussian integers are known as Gaussian primes. Ten times the integers produces multiples of ten along the x axis. Gaussian integers Z [i]. of units for the Gaussian integers modulo n, for many n. They reduced the problem to the prime power case and the three cases for various primes: 2, those congruent . View 6gaussian.pdf from CS MISC at Southern University of Chile.

Hence and is definitely a solution, and since you asked for "the" two Gaussian integers, not "all" solutions, is probably the simplest one to use. We can . Let p be an integer prime for which there is an element a in Z with a^2 + 1 = p. Write down a factorisation of. The set forms a two-dimensional lattice in the complex plane.

If , then is a Gaussian prime iff is an ordinary prime and ..

Pages in category "Units of Gaussian Integers" The following 3 pages are in this category, out of 3 total. Transcribed Image Text: Definition 10.3. A Gaussian integer is a complex number such that the real part is a real integer and the imaginary part is a real integer multiplied by the imaginary unit . A Gaussian prime is a non-unit Gaussian integer divisible only by its associates and by the units ( ), and by no other Gaussian integers. For instance, 5 is a prime number among the integers, but it can be factored into (2+i)(2-i) over the Gaussian integers. Let R be an integral domain. We know that addition is associative, and the identity element in addition is the number which when added "does nothing": $a + 0 = 0 + a = a$ for all $a \in F_p^+$. With this in mind, we are ready to de ne the notion of a prime for the Gaussian integers. The associates of are known as units: . The gaussian integers form a commutative ring.

For example, with 23 + 41i we compute the product (23 + 41i) (23 - 41i) = 2210. Here is the complex conjugate of . Moreov er, we examine the existence of solutions for the equation k ( ) = q ( ) + L , where k is a p ositive integer, L is a xed in teger, In the integers the units are +1 and -1. Find the square that contains 17+24i. In the ring of polynomials over a field F[x] the units are the non-zero constant polynomials.

Let p be a . Share The multiplicative group of the Gaussian integers is {1 . The only part that is not, perhaps, obvious is that the inverse of a This system is the most common of the several electromagnetic unit systems based on cgs (centimetre-gram-second) units.

Taken together this implies that . MHB Site Helper.

First, divide out the GCD of a and b to form a reduced Gaussian integer. Otherwise, it is called composite. Lemma 2. Here is the complex conjugate of .

Last Post; Aug 15, 2022; Replies 26 Views 413. Other Math. find all the units in the ring Z [i] of Gaussian integers. Theorem: In the Gaussian integers R, each non-zero, non-unit member can be factorized into a product of primes; furthermore, the factorization is unique up to ordering and associates. Let R = Z [ i] = { a + i b a, b Z } be the ring of Gaussian integers. The Gaussian integers modulo is the set {a + hi : a, b Zn and i2 1} and is denoted Zn[i]. Note that, in general, zand its associates are distinct from z and its associates.

So I know that Gauss integers are a+bi \ \ \ a,b \in \mathbb {Z} [/tex] Now, of course -1, 1, i. This can happen only if a2 = 1 and b2 = 0 or a2 = 0 . 1, i, and ? Two Gaussian integers and are associates if there is a unit . (a)Show that the sum or product of two Gaussian integers is again a Gaussian integer. Clearly, multiplying by a unit does not change primality. Gauss showed that (\mathbb {Z}/n\mathbb {Z})^* (Z/nZ) is a cyclic group if and only if n=1,2,4,p,p^k n = 1,2,4,p,pk or 2p^k 2pk, where p p is an odd prime and k > 0 k > 0. The exceptions to this rule occur when the real and imaginary parts of zhave the same absolute value, and when . Connect the dots to produce an infinite checkerboard, where each square is ten units on a side. Gaussian Integers. unit Gaussian integers, or units, and two Gaussian integers are called associates if they can be obtained from one another by multiplication by units. The rst step is to dene the norm of a Gaussian Integer, although it is the same as for complex numbers in general. 6 Gaussian Integers and other Rings One way that Euler, Lagrange, Jacobi, Kummer and others tackled Fermat's Last Theorem was to try Math; Algebra; Algebra questions and answers; Determine all units in the ring Z[i] of Gaussian Integers. The units of Z[i] are 1 and i.

ring theory 7 | units of gaussian ring | csir net | gate | tifr | isi | nbhm Two Gaussian integers and are associates if there is a unit usuch that = u.

Gaussian integers. The units of Z are 1.
1 Units in Z[i] An element x = a + bi Z[i], a, b Z is a unit if there exists y = c + di Z[i] such that xy = 1. A Gaussian prime is an element of that cannot be expressed as . David Joyce First we will show that this ring shares an important property with the ring of integers: every element can be factored into a product of nitely many "primes". A gaussian number is a number of the form z= x+ iy (x;y2Q): If x;y2Z we say that zis a gaussian integer. It forms a commutative ring with multiplicative identity 1 under 1 = |x|2|y|2 = (a2 + b2)(c2 + d2) But a2, b2, c2, d2 are non-negative integers, so we must have. De nition 6. If , then is a Gaussian prime iff is an ordinary prime and .. 3. In this part, you may assume any facts about the factorisation theory of Z[i], the ring of Gaussian Integers, and of Z provided that you state clearly the properties that you are using. is a Euclidean domain. Number Theory Units of the Gaussian Integers, Z[i] Thread starter Peter; Start date Oct 11, 2014; Oct 11, 2014. [1] This integral domain is a particular case of a commutative ring of quadratic integers. Units of Gaussian integers. The Gaussian primes fall into one of three categories: Gaussian integers with imaginary part zero and a prime real part with a real prime satisfying (numbers of A002145 multiplied by or ). Proof. Special Gaussian primes are and primes with and . Thread starter #1 Peter Well-known member. This article formalizes some definitions about Gaussian integers, and proves that the Gaussian rational number field and a quotient field of theGaussian integer ring are isomorphic. Now, follow the method of factoring integers over the Gaussian primes outlined in this article. When plotted in the complex plane, they form a square lattice, as shown in the left-hand diagram below. This choice of a defined quantity will make the Gaussian unit of charge

For example, 3 is itself a prime, so 3 = 3 is a prime factorization. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. Every nonzero Gaussian integer , where and are ordinary integers and can be expressed uniquely as the product of a unit and powers of special Gaussian primes. In particular, induction on the norm (not on the Gaussian [3] Euclidean division Visualization of maximal distance to some Gaussian integer 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. By Corollary 6.13, it is therefore a unique factorization domain, so any Gaussian integer can be factored into irreducible Gaussian integers from a distinguished set, which is unique up to reordering.In this section, we look at the factorization of Gaussian integers in more detail. The norm of a Gaussian integer x + iy is defined to be N(x + iy) = X2 + y2. Denition. MHB Let Z = Set of Integers. We call these four numbers the Gaussian units. We will see this is not the case. Jun 22, 2012 The Gaussian divisors of an integer display an interesting symmetric configuration. Then a Euclidean function is a func- tion from the set of nonzero elements of R to the nonnegative integers such that, for all nonzero a, b e R, we have 1. () < (ab); and 2. there exist q,r e R such that a = bq +r, and either r = 0 or (r) < (b). In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. Suggested for: Units of the Gaussian Integers, Z[i] I Primes -- Probability that the sum of two random integers is Prime.

The usual symbol for the ring of Gaussian integers is , but and [1] have also been used. Therefore the set of Gaussian integers satisfies the condition for an integral domain. 1 = a2 + b2 = c2 + d2.

The sum, difference, and product of two Gaussian integers are Gaussian integers, but only if there is an such that (1) (Shanks 1993). If p is a prime number and p 1 (mod 4), then p = a + b for some a,b . Purely real integers may be considered Gaussian integers having an imaginary part of 0. Summary Gaussian integer is one of basic algebraic integers. The norm of a Gaussian integer = a+biis de ned by N( ) = = a2 +b2. To address the OP's question, if and then for the first step of the euclidean algorithm, we seek and satisfying such that either or the norm of is strictly smaller than . Units are 1, , -1, . Since division of complex numbers is messier than in the integers, any given example of the Eu-clidean algorithm takes much longer. The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers in the set. March 3, 2022 by admin.

So, for example, N(5711i) = 3370. is a Gaussian prime since is an ordinary prime. We say that a Gaussian integer z with N(z) > 1 is . The Gaussian solution to breaking the cycle is to make 4 a defined quantity, specifically: 4 L 1 4 (Gaussian units) In fact, the symbol " 4" isn't even ever written out in any equations that involve Gaussian unitsyou'll only see factors of 4 instead. We will first describe the distinguished irreducibles we will use for Gaussian integers. . There are, however, infinitely many solutions. The most general definition of prime in a ring R is: Def: p is prime if p=ab implies either a or b is a unit. In the rst case we obtain Gaussian integers 1 Units in Z[i] An element x= a+ bi2Z[i];a;b2Z is a unit if there exists y= c+ di2Z[i] such that xy= 1:This implies 1 = jxj 2jyj= (a2 + b2)(c2 + d2) But a 2;b 2;c;d are non-negative integers, so we must have 1 = a 2+ b = c2 + d2: This can happen only if a2 = 1 and b 2= 0 or a = 0 and b2 = 1. Define a norm function on the Gaussian integers by d(a+bi)=a 2 +b 2 (the square of the modulus).

It is very easy to verify this for n=1,2 n = 1,2 and 4 4, as one can simply list out all positive integers less than and co-prime to n n. The Gaussian integers, denoted , are the numbers where are ordinary integers. A Gaussian integer is called prime if it is not equal to a product of two non-unit Gaussian integers. We define the norm N: Z [ i] Z by sending = a + i b to N ( ) = = a 2 + b 2. Gaussian integers, not both zero, then we dene gcd(w,z) to be any common divisor .

Two Gaussian integers v, w are associates if v = uw where u is a unit. Examples 6.11. So, basically a Gaussian integer is a complex number of the form a + bi, where a and b are both integers. By complex division, . Find units in a ring constituted by Gauss integers. Therefore, to restate (1), a Gaussian integer a + bi (a, b =A 0) is a G-prime if and only if N(a + bi) is a prime. The Gaussian integers Z[i] are all complex numbers a+ biwhere aand bare integers. KEITH CONRAD. The proof that there are indeed just four units in the Gaussian Integers uses a simple proof by contradiction. In such a integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. GAUSSIAN INTEGERS HUNG HO Abstract. Now a, b, c, d are all integers, so a 2 + b 2 and c 2 + d 2 must both be nonnegative integers, which must both equal exactly 1 and no greater in order to multiply to 1 in the integers. The ring of Gaussian integers is defined in the following ways: It is the subring generated by the ring of rational integers and the element (a square root of -1) in the field of complex numbers. Contributed by: Izidor Hafner (March 2011) Open content licensed under CC BY-NC-SA Snapshots Details De nition 6. For any element we consider the four numbers as associates. The example we will look at in this handout is the Gaussian integers: Z[i] = {a + bi : a, b Z}.

In particular, induction on the norm (not on the Gaussian Last Post; Jun 23, 2020; Replies 5 Views 1K. Gaussian Integers 12.1 Gaussian Numbers De nition 12.1. if a Gaussian prime divides zw then divides z or divides w. Fermat's Two Square Theorem. Knowing a Gaussian integer up to multiplication by a unit is analogous to knowing an integer up to its sign. laws hold; there is a unit (certain Gaussian integers divide every Gaussian integer; these are called units).

Knowing a Gaussian integer up to multiplication by a unit is analogous to knowing an integer up to its sign. We perform Proof. It is the integral extension of the ring of rational integers , with the image of the indeterminate denoted as . Gaussian primes are Gaussian integers satisfying one of the following properties.. 1.

This result is the key to all the remaining concepts in Units of Gaussian integers Author: Lucy Foss Date: 2022-07-09 Solution 2: The speed of light in what you would normally call Gaussian units is precisely 29,979,245,800 cm/s.

2+i and 2-i are Gaussian primes. a is called the 'real part' and b is called the 'imaginary part.' The set fa + bi : a;b are integersgare known as the 'Gaussian integers.' Gaussian integers can be visualized as points in the plane: Jacob Richey and Carl de Marcken (UW) Math Circle 3/26/2020 3/12

(explain who to get 1,-1 and i, -i) Question: Determine all units in the ring Z[i] of Gaussian Integers. Details. Factorization over Gaussian primes is unique up to multiplication by units, of which there are four: 1, -1, i .

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