Artin–Schreier theory
Okay, let's try to explain the Artin-Schreier theory in a way that a 5-year-old would understand it.
So first, let's imagine that you have a box of crayons. You know how to count them, and you know what colors they are. But what if I told you to make a new color that's not in the box? How can you do that?
Well, one way is to cheat a little bit. You can make a new color by saying that two colors are the same when they're actually different. For example, you can say that red and blue are the same color. Then, you have a new color that's a mix of red and blue!
Now, let's imagine that instead of crayons, we're dealing with numbers. Specifically, we're dealing with numbers that are roots of polynomials. This sounds complicated, but it's really just a fancy way of saying that we're looking for numbers that make a certain equation true.
Now, what if I asked you to find a number that solves the equation x^2 = -1? You might say that there's no such number, since you can't take the square root of a negative number. But what if we cheat a little bit again?
We can say that there is a number that solves this equation, and we'll call it i. Then, we can say that i^2 = -1. Now we have a new number that didn't exist before!
But wait, there's more! The Artin-Schreier theory tells us that we can use this trick to make even more new numbers. We just have to add a little twist. Instead of saying that two colors are the same, we say that two numbers are the same when they differ by a certain amount.
For example, we can say that two numbers are the same if their difference is equal to i. Then we have even more new numbers to play with!
This might all seem a little abstract and weird, but the Artin-Schreier theory is actually really important in mathematics. It helps us understand how to extend number systems in creative ways, and it has applications in cryptography and coding theory. Plus, it's always fun to make new colors – I mean, numbers!
So first, let's imagine that you have a box of crayons. You know how to count them, and you know what colors they are. But what if I told you to make a new color that's not in the box? How can you do that?
Well, one way is to cheat a little bit. You can make a new color by saying that two colors are the same when they're actually different. For example, you can say that red and blue are the same color. Then, you have a new color that's a mix of red and blue!
Now, let's imagine that instead of crayons, we're dealing with numbers. Specifically, we're dealing with numbers that are roots of polynomials. This sounds complicated, but it's really just a fancy way of saying that we're looking for numbers that make a certain equation true.
Now, what if I asked you to find a number that solves the equation x^2 = -1? You might say that there's no such number, since you can't take the square root of a negative number. But what if we cheat a little bit again?
We can say that there is a number that solves this equation, and we'll call it i. Then, we can say that i^2 = -1. Now we have a new number that didn't exist before!
But wait, there's more! The Artin-Schreier theory tells us that we can use this trick to make even more new numbers. We just have to add a little twist. Instead of saying that two colors are the same, we say that two numbers are the same when they differ by a certain amount.
For example, we can say that two numbers are the same if their difference is equal to i. Then we have even more new numbers to play with!
This might all seem a little abstract and weird, but the Artin-Schreier theory is actually really important in mathematics. It helps us understand how to extend number systems in creative ways, and it has applications in cryptography and coding theory. Plus, it's always fun to make new colors – I mean, numbers!