In 1825, German mathematician Carl F. Gauss published a paper introducing complex numbers of the form *m* + *no***i**, Where *m* and *no* are integers and **i** = (-1)^{1/2}, when investigating issues related to the reciprocity of the quadrangle. The Laws of Reciprocity represent one of the most interesting results of Number Theory. These laws were born from the Quadratic Reciprocity Theorem which was demonstrated by Gauss and previously conjectured by Pierre de Fermat, Leonard Euler and Joseph Legendre. David Hilbert, and later André Weil, have generalized these laws and are not yet fully understood in more general situations.

Probably the Quadratic Reciprocity Law (LRQ) was one of the first profound results of modern Number Theory. Originally, it was independently conjectured by Euler and Legendre in the first half of the 18th century. However, they only obtained the demonstration for particular cases. In 1795 Gauss discovered it for himself, but did not feel that he could demonstrate it, and in a letter reported that the demonstration tormented him for a year and consumed his best efforts. At nineteen years old, on April 8, 1796, Gauss gave the first demonstration of the Quadratic Reciprocity Law and during his lifetime found other demonstrations of this result.

Before we state this result, let us recall the concept of congruence seen in the last columns on the “Riemann Zeta Function and the Internet”. Gauss introduced the concept of congruence in the first chapter of his work “Disquisitiones Arithmeticae” published in 1801. At that time he also introduced the notation “≡” which made this concept a powerful technique in Algebra and Number Theory. Let's go to the definitions.

We consider two integers *The*, *B* and *no* a positive integer. If *no* divide *The* - *B* we say that

*The* é **congruent** The *B* **module** *no*, and we wrote *The* ≡ *B* (mod *no*).

For example: 27 ≡ 2 (mod 5), because 5 divides 27 - 2 = 25, 7 ≡ 7 (mod 4), because 4 divides 7 - 7 = 0.

Therefore, *The* ≡ *B* (mod *no*) it means that *no* divide *The* - *B*; soon there is an integer *k* such that *The* - *B* = *kn* by the definition of divisibility. For example, 37 ≡ 2 (mod 5) because 37 - 2 = 35 = 7 • 5. Given the integers *The* and *no* we know from the Division Algorithm that there are integers *what* and *r* respectively referred to as quotient and remainder such that: *The* = *when* + *r*where 0 ≤ *r* < *no*; soon *The* - *r* = *when*, that is, *no* divide *The* - *r*. Therefore, by the definition of congruence *The* ≡ *r* (mod *no*). The rest *r* can assume any value between 0 and *no* - 1, so we conclude that every integer *The* is congruent module *no* to exactly one of the values between 0, 1, 2,…, *no *- 1. The set {0, 1, 2,…, *no *-1} of *no* integers that are the remains of module divisions *no*, is called the module waste class *no*. If we fix *no* = 7, then module class 7 has exactly 7 elements, namely: 0, 1, 2,…, 6. Therefore, whatever the integer is, it is congruent with a single element of module 7 class. For example, 20 is represented by 6 in the waste class, as 20 ≡ 6 (mod 7).

Due to the many similar properties that congruences and equality satisfy, Gauss chose the symbol “≡” for the congruence sign. Notice that *The* ≡ *The* (mod *no*) what if *The* ≡ *B *(mod *no*), then *B* ≡ *The *(mod *no*). The addition, multiplication and potentiation operations behave as follows: if *The* ≡ *B* (mod *no*) and *ç* ≡ *d* (mod *no*), then: *a + c * ≡ *b + d* (mod *no*), *The *•* ç * ≡ *B *•* d* (mod *no*), *The ^{r}* ≡

*B*(mod

^{r}*no*).

Euler wondered under what conditions the congruence *x*^{2} ≡ *what* (*mod* *P*) admitted solution to cousins *P *and* what* Dice*. *When this congruence has a solution we say that *what* it is a *quadratic residue* *module* *P*. Otherwise, we say that *what* it is a *non quadratic residue* *module* *P*. Therefore, the *quadratic waste* *module* *P* are those elements of the module residue class set *P* which are square. If we fix *no* = 7 then the class of module modulus 7 has exactly 7 elements, namely: 0, 1, 2,…, 6, and exactly 3 elements that are square, namely: 1 = 1^{2}, 4 = 2^{2}, 2 = 3^{2}, that is, 3^{2} = 9 ≡ 2 (mod 7). Therefore, integer 2 is quadratic residue modulus 7. However, 5 is non-quadratic residue modulus 7, since none of the elements of the set {1, 2, 3, 4, 5, 6} satisfy the equation. *x*^{2} ≡ 5 (*mod* 7).

The interest of the Quadratic Residue Theory lies in the following question: for any odd prime numbers *P* and *what*, there is a relationship between the ownership of *P* be module quadratic residue *what *with the property of *what *be module quadratic residue *P*? Therefore, we are discussing the nature of reciprocity of quadratic residues.

In 1640 Fermat enunciated the following theorem, now known as Fermat's little theorem:

“If *P* is an odd cousin that doesn't divide an integer *The*, then *The ^{P}*

^{ - 1}≡ 1 (mod

*P*).”

How *P* is odd, it follows that (*P* - 1) / 2 is an integer, so we have to: *The ^{(P}*

^{ - 1)/2}≡ 1 (mod

*P*).

Now known as the Euler Criterion, this was the starting point for Euler to investigate a LRQ demonstration. Let us state the Euler Criterion:

“*Let p be an odd prime and an integer such that p does not divide the*.

*The number a is quadratic remainder module p if, and only if*, *The ^{(P}*

^{ - 1)/2}≡ 1 (mod

*P*).”

For example, *The* = 3 is a non quadratic residual modulus *P *= 7, because 3^{3} = 27 ≡ -1 (mod 7).

On the other hand, *The* = 3 is a modulus quadratic residue *P *= 11, because 3^{5} = 243 ≡ 1 (mod 11).

However, this criterion is not practical. For example, if we want to decide if integer 17 is a quadratic residue module 1987, we need to decide if 17^{993} is congruent to 1 module 1987 (note that (1987-1) / 2 = 993). Therefore, there is a need to investigate whether there is a more convenient method.

Euler focused on the situation where both integers *P* and *what* they are positive, odd and distinct prime numbers. Legendre attempted to give a demonstration of this fact in 1785, but he assumed a result whose demonstration is much deeper than the LRQ demonstration, namely that certain arithmetic progressions contained infinite prime numbers between their elements.

However, Legendre introduced the following symbol (*The*/*P*): (*The*/*P*) = 1 if *what* is a quadratic residue of *P*, and (*The*/*P*) = -1, otherwise. This symbol (*The*/*P*) satisfies many interesting properties. For example, if *P* is an odd cousin and *The*, *B* are integers not divisible by cousin *P*then: the symbol is multiplicative, ie ((*ab*)/*P*) = (*The*/*P*) (*B*/*P*); if *The* ≡ *B *(mod *P*), So (*The*/*P*) = (*B*/*P*).

With this symbol (*The*/*P*), known as the Legendre symbol, the LRQ is conveniently expressed as follows:

(*what*/*P*) (*P*/*what*) = (-1)^{(P - 1) / 2. (what - 1) / 2}.

LRQ can be formulated in other ways. Multiplying the above equality by (*P*/*what*) we get equality

(*what*/*P*) = (-1)^{(P - 1) / 2.(what - 1) / 2}(*P*/*what*),

because (*P*/*what*) = ± 1. Let's decide if integer 30 is a quadratic residue modulo 53 using the LRQ. We first note that:

(15/53) = (3/53)(5/53).

(3/53) = (-1) ^{(3 - 1)/2. (53 - 1)/2 } (53/3) = (53/3) = (2/3),

for the rest of the division of 53 by 3 is 2, that is 53 ≡ 2 (*mod* 3). Since 2 is a non-quadratic module 3 residue, it follows that (2/3) = -1. By the LRQ, (5/53) = (-1) ^{(5 - 1)/2. (53 - 1)/2 } (53/5) = (53/5) = (3/5), because the rest of the division of 53 by 5 is 3, that is, 53 ≡ 3 (*mod* 5). As 3 is non-quadratic residue modulus 5, it follows that (3/5) = -1. Therefore, (15/53) = (3/53) (5/53) = (-1). (-1) = 1 implies that 15 is quadratic residue modulus 53.

Gauss is considered by many to be one of the three greatest mathematicians in history, alongside Archimedes and Newton. At seventeen he decided to correct and complete the research his predecessors had developed in arithmetic. Gauss had a keen interest in arithmetic questions and his phrase is known:

*“Mathematics is the queen of science and arithmetic is the queen of mathematics*.*”*

Gauss's work is a source of inspiration for his creativity and a deep and modern look at mathematical questions. In his book “Disquisitiones Arithmeticae” he studies equations of the type *x ^{no}* º

*The*(mod

*P*). This is a difficult problem that still requires investigation. However, by studying the situation in which

*no*= 2, discovered and demonstrated the LRQ.

In the period between 1808 and 1832, Gauss continued to investigate similar laws for powers higher than squares, that is, relations between *P* and *what* such that *what* were a cubic rest of *P*, (*x*^{3} º *what* (*mod* *P*)), or bikadratic residue (*x*^{4} º *what*(*mod* *P*)), and so on. During this investigation, Gauss made some discoveries and realized that the investigation became simpler by working on complex numbers.* m* + *no*i where *m* and *no* are integers and **i** = (-1)^{1/2}.

Gauss developed a prime factorization theory for these complex numbers Z**i** currently known as *Gaussian Integers* or *Gaussian Integers* in honor of him.

Gauss demonstrated that the set of Gaussian integers, provided with the addition and multiplication operations, gives rise to a structure called the integrity domain. In addition, Gaussian integers admit a prime decomposition, this decomposition is unique unless of the order of factors exactly as with the whole number set.

Gauss generalized the idea of integer when defining the set Z**i**. He found that much of Euclid's old theory of integer factorization could be carried to the Z domain.**i** with important consequences for Number Theory. However, divisibility issues become complex in this domain. Note that 5 is a prime number in Z, but no longer prime in Z**i**. Indeed,

(1 + 2**i**).(1 - 2**i**) = 1 - 2**i** + 2**i** - 4**i**^{2} = 1 - 4.(-1) = 5.

A natural question arises: what are the prime numbers of the Z health domain**i**?

This and other questions regarding Gaussian integer arithmetic will be commented in our next column.

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