What are some practical examples of ring and field arithmetic?

Question

Mathematics

Hello! Now I study algebra, and specifically finite fields and rings. And all examples and exercises are abstract. And what are the practical applications of these arithmetic? At school, all the tasks were about trains or pies, but are there any here?

Answer 1

There are just a bunch of applications in cryptography. Any RSA, DH, elliptic curves - it's all based on the properties of some special finite fields.
But I like this puzzle: there is a table of light bulbs-buttons nx m. Some are on fire, some are not. You can click on a light bulb and it will switch. But along with it, 4 neighboring lamps will also switch (if you pressed the corner button, then only 2). It is necessary to extinguish all the lamps in the minimum number of clicks. How to solve it in general without full or partial enumeration? It is important to note that it is pointless to click on the lamp 2 times, because these 2 clicks will simply cancel each other. It doesn't matter in what order you press the lamps. The end result will be the same.
And then the math kicks in! Let's enter the variables x_ij - how many times we click on the light bulb in positions i, j. These variables are field elements modulo 2. Because clicking the button 2 times is the same as clicking 0 times. We make linear equations that the sum of pressing all the buttons that affect a given lamp gives 0 or 1 modulo 2 (depending on whether this lamp is initially lit).
And then this system of equations can be simply solved by the Gauss method. Why? After all, he works with real numbers? But no! Gauss doesn't care what field to work on. We do all calculations modulo 2 - and we get a solution in the form of 0 and 1 for all variables.
Fields modulo prime numbers can still be used, for example, to implement a fast Fourier transform for fast multiplication of long numbers without using float operations, which is required in the standard implementation (it generally works with complex numbers). And this implementation is faster and more accurate.

Answer 2

Cryptography. Bitcoin is built on elliptic curve arithmetic over finite fields.

Answer 3

Abstractions of this level have no practical applications, but they allow us to simplify simpler tasks. The standard application in practice looks like this:
1. Consider a practical number system.
2. Proves that it is a group.
3. Now we can use ready-made properties and formulas for our specific number system. That is, we skipped several dozen pages of formulas for our problem.