Context: Take the set of numbers module p, a prime number. Most of the field axioms (addition is a group, multiplication is a semi-group, the distributive law) are easily verifiable as an exercise (it’s a simple restatement of theorems about numbers). The big one is proving that for every number not 0, there is an inverse.
For some reason, most people will resort to the Euclidean algorithm. But there’s a simpler way.
- Prove that there are no zero divisors. This is a different way of saying that if a prime number divides a product, it divides one of the terms.
- Prove that this implies ax=bx implies a=b or x=0
- Therefore, the function x->ax is one-to-one, on a finite set.
- Therefore, it is onto.
- Therefore, 1 is in the image. Or, there exists an a, ax=1.