kowalski said:i. Do you mean that there are infinite "real numbers, say R_p" (one for each prime p) apart from the usual reals R?.
Why are they interesting, what propierties does they have?.
ii. Can you give a (no too esoteric) example of the use of p-adics in Number Theory?. Do they solve 'real' problems, or problems originated in their own existence?.
iii. Any applications out of Number Theory? (math or physics?).
You could start with the Hasse-Minkowsky Theorem.kowalski said:ii. Can you give a (no too esoteric) example of the use of p-adics in Number Theory?.
P-adic numbers are a type of number that extend the concept of the real numbers. They were first introduced by mathematician Kurt Hensel in the early 20th century and are based on the idea of p-adic valuation, where p is a prime number.
P-adic numbers differ from real numbers in several ways. One of the main differences is that p-adic numbers are based on a different metric, called the p-adic metric, which is used to measure the distance between numbers. This metric leads to some unique properties of p-adic numbers, such as the fact that they can have infinitely many decimal places to the left of the decimal point.
P-adic numbers have various applications in mathematics, physics, and computer science. They are used in number theory, algebraic geometry, and cryptography. In physics, p-adic numbers have been used to study string theory and quantum mechanics. They are also used in computer science for error-correcting codes and in algorithms for solving problems in finite fields.
The arithmetic operations with p-adic numbers are very similar to those with real numbers. Addition, subtraction, and multiplication are carried out in the same way as with real numbers. Division, however, is defined differently due to the unique properties of p-adic numbers. It involves finding a reciprocal, which can be done using the p-adic valuation. Some basic rules for arithmetic with p-adic numbers include: adding or subtracting two p-adic numbers results in a p-adic number, multiplying two p-adic numbers results in a p-adic number, and dividing two p-adic numbers results in a p-adic number if the divisor is not divisible by the prime p.
There are many resources available for learning more about p-adic numbers. Some suggestions include reading textbooks on number theory or algebraic geometry, taking a course in number theory or abstract algebra, or attending lectures or seminars on p-adic numbers. Additionally, there are many online resources, such as videos and articles, that can provide an introduction to p-adic numbers. It may also be helpful to consult with a mathematician or join a math forum to ask questions and discuss p-adic numbers with others.