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Gaussian integer arithmetic


The work of the German mathematician Carl F. Gauss is universal. Gauss produced with ease in all branches of mathematics. He even made important contributions in astronomy, developing a method of calculating orbits of celestial bodies from a small number of observations. To this day, this method is used to track satellite orbits. However, the pleasure I felt for research in arithmetic is notorious. His monumental work “Disquisitiones Arithmeticae” laid the foundations of the modern Number Theory.

In 1825, he published a paper introducing complex numbers as follows. The + Bi, Where The and B are integers and i = (-1)1/2. This set is indicated by Zi and is called Gaussian Integers or set of Gaussian Integers in honor of their creator.

Gauss investigated issues related to bikadratic reciprocity, ie relations between prime numbers P and what, such that the cousin what were a cousin's remainder of cousin P, x4 = what(mod P)when he realized that research was becoming simpler by working on Zi. Thus Gauss extended the idea of ​​integer when defining Zibecause he discovered that much of Euclid's old theory of integer factorization could be carried over to this set with important consequences for Number Theory.

This generalization of the whole number set gives special examples of much deeper developments that we call Algebraic Number Theory. This theory is deep and powerful. In addition to its interest and fascination for its own properties, it provides many applications to Number Theory that allow an understanding of various previously obscure and mysterious phenomena. For example, we consider far more general algebraic irrationalities, ie roots of algebraic equations of all degrees that are beyond quadratic irrationalities.

Let's discuss some of the arithmetic properties of Gaussian integers. First, we observe that Zi is a subset of C, the set of complex numbers. Therefore, consider the set Zi with the addition and multiplication operations inherited from C. That is, if z1 = The + iB and z 2 = The + iB So

z 1 + z 2 = (The + ç) + i(B + d)

and

z 1 . z 2 = (The + ç) + i(B + d).

The neutral element of the addition is 0 = 0 + 0i, the neutral element of multiplication is 1 = 1 + 0i and finally -1 = -1 + 0i. All other properties, such as associative addition and multiplication, commutative addition and multiplication, distributive, are inherited from C. Note that for every integer no we have the ID no = no + 0i, or yet, no = no. Therefore, 0 = 0, ±1 = ±1, ±2 = ±2,…

Divisibility issues become complex in this set. Note that the integer 5 is prime in Z. However, in Zi we have

(1 + 2i).(1 - 2i) = 1 - 2<>

i + 2i - 4i2 = 1 - 4(-1) = 5.

Since not every integer prime is a Gaussian prime, some questions naturally arise: What are the prime numbers of this ring? Are there infinite Gaussian cousins? Is it possible to decompose Gaussian integers into prime factors in a single way unless of the order?

To comment on these issues, which involve the notion of divisibility in Zi, we need to define what is divisibility in Zi.

Suppose that x and y are distinct Gaussian integers, where y ¹ 0. We say that y divide x, and we indicate by y çxif there is a gaussian integer w such that x = wy. For example,

(1 + i) ç2, because 2 = (1 + i)(1 - i)

and

(1 + i) ç(1 - i) because 1 + i = i(1 - i).

Now notice that 1 + 2i don't divide 1 - i. Otherwise we would have 1 + 2i = (ç + di)(1 - i) Where ç and d belong to Z. We get 1 + 2i = ç + d + (d - ç)i, that is, ç + d = 1 and d - ç = 2 equaling, respectively, the real part and the imaginary part. Adding up the two previous equations we get 2d = 3. However, d It's an integer!

Will be the definition of divisibility in Zi compatible with the definition of divisibility in Z? We want to know, for example, if it is possible 3 to divide 7 into Zi. The answer could not be more significant:

there is compatibility between the definition of divisibility

given for Gaussian integers relative to the definition given for the integers.

In fact, suppose that x and y, y ¹ 0, are elements of Z such that y çx in Zi. So there is w = ç + di in Zi such that x = wy, that is, x = (ç + di)y = cy + dyi. Soon, x = cy and 0 = dy. How y ¹ 0, 0 = dy implies that d = 0 and thus w = ç It's an integer! Therefore, x = wy = cy. We conclude that if y çx in Zi, then y çx in Z.

We know that 1 and -1 divide all integers. Similarly, it is shown that ± 1 and ± i divide all Gaussian integers. Therefore, ± 1 and ± i are called units of Gaussian integers. If w is a unit of the gaussian integers and x and y are Gaussian integers such that x = wy, then we say that x and y are associated elements. Note that, 1 + i and 1 - i are associated elements because 1 + i = i (1 - i).

We are now able to define Gaussian cousins: a gaussian integer x is a gaussian cousin if the only dividers of x are their associates and the units of Zi. For example, the integer 2 not a cousin in Zi, because

<>

i(1 - <>

i)2 = i(1 - 2i + i2) = i(-2i) = -2i2 = <>

2.

As noted earlier, there are many properties that Gaussian integers and integers have in common. We know from previous columns that there are infinite prime integers of form 4k + 3. In turn, it is shown that every prime integer of form 4k + 3 is a gaussian cousin! So there are infinite Gaussian cousins. Gaussian cousins ​​are shown to be precisely:

The whole gaussian 1+ i and their associates; the prime integers of the form 4k + 3 and its members; and the numbers to ± Biwhere the2 + b2 is a prime integer of the form 4k +1,

and its associates.

We observe that the associates of a Gaussian integer x are obtained by multiplying x by ± 1 or ± i.

If P = 3 So P = 3 = 4.0 + 3; soon the whole gaussian 3 He is a Gaussian cousin. If P = 5, then P = 5 = 4.1 + 1 implies that 2 + i and 2 - i and their associates are Gaussian cousins.

Like every prime integer or is of form 4k + 1 or form 4k + 3, we conclude that there are two Gaussian cousins ​​corresponding to each prime integer of form 4k + 1, and a Gaussian prime that corresponds to each prime integer of form 4k + 3. Thus, every Gaussian prime is a factor of a single prime integer. We often say that the cousins ​​of form 4k + 3 remain prime at Zithat the cousins ​​of form 4k + 1 decompose in Zi, is that 2 = -i(1 + i) branches into Zi.

We have observed that until now we have no elements to compare Gaussian integers by the known order relation “<”. Let's assume that this definition can be extended to Gaussian integers. We know that whatever the extended definition we will always have to 0 < 1. How i ¹ 0, if we suppose i < 0, so necessarily 0 < -i and therefore, 0 < (-i)2 = -1, which is false! On the other hand, if we suppose 0 < i, then 0 < i2 = -1, which is also false!

To compare Gaussian integers we can define a rule function with domain in them that assumes values ​​in the N naturals. Therefore, we define N of a gaussian integer x = The + Bi, per N(x) = N(The + Bi) = The2 + B2. The norm plays an important role because, as we know, inequalities are fundamental in the study of the arithmetic and algebraic properties of integers.

In Zi a division with remainder is very similar to the Euclidean division defined in integers:

Be x and y gaussian integers, with y ¹ 0. So there are Gaussian integers w and z

such that: x = wy + z, with N(z) <N(w).

So the division with remainder in Zi It is algorithmic. This fact allows us to calculate the greatest common divisor of two non-null Gaussian integers.

Integers satisfy a very important property in Number Theory: single factorization, that is, any positive integer is expressed in a unique way, unless of the order of factors, as the product of prime numbers. Gaussian integers also satisfy this important arithmetic property, that is, they allow for prime decomposition, and this decomposition is unique less than the order of the factors.

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