# Diophantine Equations V

## The birth of Algebraic Number Theory

Since the seventeenth century, great mathematicians have tried to reconstruct the wonderful demonstration that Fermat claimed to possess for the fact that has no solution to positive integers when no > 2. But such a demonstration did not fit that margin of his copy of Bachet's work: "Arithmetica de Diophantus", a work consisting of what was left of Diophantus's work. It is reported that in 1742, Euler, the greatest mathematician of the 18th century, asked his friend Clerot to search Fermat's house for some piece of paper with any indication of Fermat's theorem demonstration, but nothing was found. However, Euler gave the first correct but incomplete demonstration for the case of the exponent no = 3. In the event that no = 4 the demonstration is attributed to Fermat and, as we noted earlier, is based on a form of induction invented by Fermat called the "Method of Infinite Descent." In 1825 Legendre and Dirichlet independently demonstrated the case. no = 5 using the "Infinite Descent Method" and in 1838 Gabriel Lamé demonstrated the case no = 7, also using the "Infinite Descent Method".

In the 19th century, French mathematics Sophie Germain assumed the identity of a man to conduct his mathematical research. She made one of the greatest advances of the century in solving Fermat's Last Theorem (UTF), finding a general result rather than demonstrating a particular positive integer. no greater than 2. She has shown that if P and 2P + 1 are both prime numbers, for example P = 3 and 2p + 1 = 2.3 + 1 = 7, so there's no whole solution x, y, z with but under the assumption that P don't divide X Y Z. As a special case, she demonstrated that if, so one of the integers x, y and z is divisible by 5.

In the mid-nineteenth century mathematicians braved other directions to demonstrate the UTF. The greatest success was achieved by Ernest Kummer. In 1843, Kummer submitted a demonstration to Dirichlet, based on an extension of integers, to include algebraic numbers, that is, numbers that satisfy polynomial equations with rational coefficients. However, Dirichlet had tried hard to give a demonstration of the theorem and thus detected a flaw in the argument: Kummer had assumed that algebraic integers allow for single decomposition in primes as with integers, but this is not true in general. Kummer didn't let himself down and returned to his investigation with redoubled effort. To ensure single factorization, in the set of algebraic integers, he invented the concept of ideal numbers. Incorporating these new entities into algebraic numbers, Kummer demonstrated the statement for a large class of prime numbers called regular cousins. Although there are infinite cousins ​​that are not regular, Kummer showed that the theorem was true for many values ​​of no. In particular, it showed that the theorem was true for all prime exponents less than 100 except for 37, 59, and 67, as these are irregular cousins. On the other hand, every time the theorem for a given exponent is demonstrated no, it is demonstrated for all multiple exponents of no and then it is sufficient to demonstrate it to prime exponents. Thus, Kummer demonstrated the theorem for all multiples of these exponents. The basic idea of ​​ideal numbers gave rise to algebraic number theory, one of the pillars of Number Theory and one of the most important branches of Abstract Algebra known as Ring Theory.

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