No problem here. First of all, I think you have remarked that the field of p-adic numbers is of positive characteristic. To create a local field easily, you can take any integral ring (in your case, take a ring R of positive characteristic like the ring of polynomials over F_p), and then LOCALIZE this ring according to some prime ideal of R (in the previous example, take some prime ideal like the ideal of polynomials multiple of X, then the localization is the ring of polynomial fractions P1/P2 where P2 is not multiple of X).zarei175 said:
A local field of positive characteristic is a mathematical concept used in fields such as algebraic geometry and number theory. It is a field with a finite characteristic, meaning that the sum of a number with itself a certain number of times will eventually equal zero. Examples of local fields of positive characteristic include finite fields and some types of p-adic fields.
Local fields of positive characteristic are important because they provide a rich and diverse mathematical landscape for researchers to explore. They have applications in many areas of mathematics, including cryptography, coding theory, and algebraic geometry. They also have connections to other fields such as physics and computer science.
An example of a local field of positive characteristic is the finite field GF(p), where p is a prime number. This field has p elements and is often used in coding theory and cryptography. Another example is the p-adic numbers, which are used in number theory and algebraic geometry.
The main difference between local fields of positive characteristic and those of zero characteristic is the behavior of addition and multiplication. In fields of zero characteristic, the characteristic is zero, meaning that the sum of any number with itself will never equal zero. In contrast, in fields of positive characteristic, the characteristic is a prime number, so the sum of any number with itself a certain number of times will eventually equal zero.
Local fields of positive characteristic have many real-world applications, such as in error-correcting codes used in communication systems, cryptography for secure communication, and algorithms for solving polynomial equations. They also have applications in physics, such as in the study of quantum mechanics and particle physics. Additionally, local fields of positive characteristic have connections to computer science, particularly in the field of computational complexity theory.
