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How did Kurt Hensel uncover the mystery of p-adic numbers in 1897?
In the field of number theory, Kurt Hensel first systematically expounded the concept of p-adic numbers in 1897. This theory has influenced many branches of mathematics to this day. As an extension of rational numbers, p-adic numbers are unique in that they are based on prime numbers and utilize a calculation method that is completely different from the conventional decimal system, thus providing mathematicians with a completely new perspective on the properties of numbers and their operations.
The emergence of p-adic numbers not only expands the concept of numbers, but also provides new methods for solving certain specific mathematical problems.
The p-adic number system based on prime number p introduced by Hensel is somewhat similar to the real numbers we are familiar with, but it is completely different in operation and structure. The representation of p-adic numbers is similar to that of decimal numbers, but their digits are based on the prime number p instead of ten, and the direction of expansion is exactly the opposite, which brings very interesting properties to calculations.
Basic concepts of p-adic numbers
P-adic numbers are an infinite sequence that expresses integers based on the base of a prime number p. For a given prime number p, a p-adic number can be written as a sequence of the form s = ∑ a_i * p^i, where each
This expression method, which is completely different from the traditional number system, allows some mathematical concepts that were once difficult to understand, such as the convergence of numbers, to find new explanations within the framework of p-adic numbers.
Hensel's theory provides the basis for understanding modular arithmetic. In short, modular arithmetic involves "approximating" every integer to the remainder of division by some positive integer n, in such a way that the approximation retains the same form throughout the operations in the number system. Hensel introduced modular arithmetic with prime numbers, which enabled him to gradually obtain solutions to certain problems through a series of simple steps.
Lemma introduced by Hensel
In the theory of p-adic numbers, two basic lemmas are of crucial importance. First, every non-zero rational number can be expressed in the form p^v * (m/n), where v, m, and n are integers, and neither m nor n is divisible by p. . Second, every rational number r can be uniquely expressed in the form r = a * p^v + s, where s is a rational number greater than v, and a is a number satisfying 0 < a An integer of type < p .
These two lemmas not only simplify the mathematical operation process, but also provide a solid foundation for the later derivation of the properties of p-adic numbers.
The establishment of these basic theories opened a new door for Kurt Hensel in his exploration of mathematics, allowing later mathematicians to conduct more in-depth research on this basis and explore the unknown world of numbers.
Application and influence of p-adic numbers
Hensel's p-adic number theory is not limited to theoretical mathematics, but has a profound impact on the calculation of arithmetic processes, the solution of equations and their applications. Mathematicians have discovered that p-adic numbers can help them solve some problems that are difficult to deal with in classical mathematics. For example, significant progress has been made in p-adic analysis, algebraic geometry, and certain screening procedures in number theory.
The development of this innovative theory not only led mathematicians to a deeper understanding of the structures represented by rational numbers, but also prompted them to rethink the role of numbers in mathematics.
As research continues to deepen, the mathematical community has gradually realized the importance of p-adic numbers. This theory plays a vital role in all areas of mathematics, especially in number theory and algebra, where their applications are becoming increasingly widespread. Today's researchers are still exploring more potential applications of p-adic number theory, which shows that p-adic numbers are still an active and open research field.
Today, Hensel's theory is not only a milestone in the history of mathematics, but also a key cornerstone for the further development of mathematical knowledge. It is in the process of our exploration of p-adic numbers that we wonder how the future of mathematics will develop and whether there will be new breakthroughs.
