In details

Gaussian Integers and the origins of Algebraic Number Theory


Between 1808 and 1825, German mathematician Carl F. Gauss investigated issues related to cubic reciprocity (x3 º what(mod P) Where P and what prime numbers) and to the two-way reciprocity (x4 º what(mod P) Where P and what prime numbers), when he realized that this investigation was made simpler by working on Zi, the ring of Gaussian integers, than in Z, the set of integers. The set Zi is formed by the complex numbers of the form The + Bi, Where The and B are integers and i = (-1)1/2.

Gauss extended the idea of ​​integer when defining the set Zibecause he discovered that much of Euclid's old theory of integer factorization could be carried to Zi with important consequences for Number Theory. He developed a Prime Factorization Theory for these complex numbers and demonstrated that this prime decomposition is unique, as with the whole number set. Gauss's use of this new type of number was of fundamental importance in demonstrating Fermat's Last Theorem.

Gaussian integers are examples of a particular type of complex number, that is, complex numbers that are solutions of a polynomial equation.

Thenoxno + Then-1 xno-1+… + The1x + The0 = 0,

where all the coefficients Theno, Then-1,… , The1, The0 are integers. These complex numbers that are roots of a polynomial equation with integer coefficients are called algebraic integers. For example, the imaginary unit, i, is an algebraic integer because it satisfies the equation x2 + 1 = 0, square root 21/2 of 2 because it satisfies the equation x2 - 2 = 0. Note that the numbers i, 21/2 are examples of algebraic integers and are not integers.

There are infinite algebraic numbers and infinite non-algebraic real numbers, such as the Euler number. and, or as the area Pof a circle of radius 1. A number that is not algebraic is called a “transcendent number”. The transcendent numbers are all irrational. However, the reciprocal is not true, since 21/2 is an irrational and algebraic number as we saw above.

The generalization of the notion of integer to algebraic integer gives special examples of much deeper developments that we call Algebraic Number Theory.

Much of the Algebraic Number Theory developed through attempts to solve the diophantine equation, better known as the Fermat Equation.

xno + yno = zno ,

because algebraic integers appear naturally as a tool to address this problem.

In the 1840s the importance of the concept of single factorization became evident. In 1847 the French mathematician Gabriel Lamé (1795-1870) announced a demonstration of Fermat's Last Theorem for every exponent. no in this Fermat equation. However, the mathematician Joseph Liouville (1809-1882), observing the proposed method, pointed out that the demonstration assumed the uniqueness of the single factorization in a subtle way. Liouville's suspicion was confirmed when he later received a letter from the brilliant German mathematician Ernest Kummer (1810-1893) showing that the uniqueness of single factorization failed in some situations. The first starting for no = 23. Kummer had published an article three years ago showing that single factorization did not work in certain situations thus destroying Lamé's demonstration. Unfortunately, Kummer's article was published in an obscure magazine and went unnoticed by Lamé.

In 1843 Kummer believed that he had demonstrated Fermat's Last Theorem using the Q-body of rational numbers, added to roots. P-ths of the unit, this is a complex number V such that VP = 1 where P is an odd prime number. Kummer considered the primitive root P-th V of the unit, this is a complex number V such that VP = 1 but Vno Quando 1 when 1 < no < P. Consider Q (V) denoting the set of all numbers of the form

Thep-2Vp-2 + Thep-1Vp-1+… + The1V + The0 = 0,

where the coefficients Thep-2, Thep-1,… , The1 and The0 They are rational numbers.

The numbers in Q (V) that have integer coefficients are called algebraic integers of Q (V). For example, the number ½ + 3V is an element of Q (V), but is not an algebraic integer; 4 - 8V + 3V2 + V3 is an algebraic integer.

Kummer observed that sums differences, products and quotients of Q elements (V) are elements of Q (V) and that the sums, differences and products of algebraic integers are algebraic integers. In this way, Kummer extended the Gaussian Integer Number Theory to the set of algebraic integers of a body. He then took the following decomposition of the Fermat equation to no = P,

xP + yP = (x + y 1) (x + y V)… (x + y Vp - 1) = zP.

So he demonstrated that this equation has no solution x, y, z with X Y Z ¹ 0. However, Kummer needed the fact that for the integers of Q (V) The property of the single factorization is valid and this fact is not generally valid. The unique factorization property is valid for P = 3, 5, 7, 11, 13, 17, 19, but not valid, for example, for P = 23. This property is not valid for an infinite number of primes. P.

Kummer had the brilliant idea of ​​creating more integers in order to regain the property of single factorization. However, these integers did not belong to Q (V). The idea was to use these new integers as factors of the algebraic integers of Q (V) in such a way that the single factorization could be recovered. These new integers were called by Kummer Ideal Numbers and considered them as follows:

(Thep-2 Vp-2 + Thep-1Vp-1+… + The1V + The0)1 / r

where the coefficients TheP -2, TheP -1,… , The1 and The0 are integers and r is a positive integer. The number r is not arbitrary, its choice is related to certain allowable values ​​according to the choice of

The = Thep-2 Vp-2 + Thep-1Vp-1+… + The1V + The0.

Continuing this line of reasoning is an integer H, called the body class number, which only depends on the given body Q (V) and is such that whatever The given, all permissible values ​​of r share H. When Q (V) has the property of the single factorization, the value r = 1 is obviously what we need to restore single factorization. This is reflected in the fact that class number H will be equal to 1 if and only if Q (V) has the property of unique factorization.

When Kummer revised his demonstration of Fermat's Last Theorem, under a new look, he realized that he could demonstrate it to more prime but not all prime exponents. He found a demonstration that was worth to cousins ​​who didn't share H, the class number associated with the bodyQ (V). He thus recognized that some cousins ​​had a pattern he called regularity: if the cousin P don't divide H It's called a regular cousin, and it's called an irregular cousin otherwise. Using this property of regularity that some prime numbers present, Kummer was able to demonstrate that Fermat's Last Theorem applies to all exponents. no = P that are regular cousins. The only irregular cousins ​​smaller than 100 are P = 37, 59, 67.

In 1850, overcoming the difficulties of uniqueness of single factorization and introducing the 'ideal' Complex Number Theory Kummer demonstrated Fermat's theorem for all exponents up to 36 and all prime exponents below 100 except for prime exponents 37, 59 and 67. It is observed that although P = 23 does not possess the unique factorization property, Kummer's result on regular cousins ​​shows that Fermat's theorem is true for this exponent. In addition, Kummer has also developed powerful methods with applications to many other mathematical problems and produced important work in atmospheric and ballistic refraction.

This theory took a different form from what Kummer bequeathed us. The mathematician Dedekind (1831-1916) reformulated the ideal number concept proposed by Kummer, proposing the fundamental key concept of the ideal of a ring that remains today. Dedekind's definition is distinct from Kummer's definition, but it is shown that they are equivalent.

In the next column, we will study some of the factorization of Dedekind's ideals.

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