Artin's criterion

Ok kiddo, have you ever heard of something called a polynomial? It's a math thing that looks kinda like this: x^2 + 3x - 4. It's made up of different powers of a letter (like x or y) and some numbers.

Now, let's say we have a really special polynomial that has whole number coefficients, meaning all the numbers in it are either whole numbers, like 1 or 2, or 0. And let's say this polynomial can't be factored into smaller pieces with whole number coefficients.

Artin's criterion says that if we can find a number (let's call it "p") that's a prime number (like 2, 3, 5...) and the polynomial can't be solved using whole numbers but it CAN be solved using fractions (we call those "rational" numbers), then that means the polynomial can actually be factored into smaller pieces with whole number coefficients!

Isn't that cool? It's like a secret code for figuring out if something that seems impossible to break apart can actually be broken apart into smaller pieces. And we use prime numbers to help us crack the code.