22.3.2.1. Proof by Contradiction

So it turns out that this is not accidental and indeed, this is the case for every finite field. So we can prove the following. That if you take any finite field F with an abstract plus and dot operation, then its characteristic is always a prime number. It cannot be a composite number and at least the theorem is true with respect to the examples that we had seen already in this lecture.

The theorem states that every finite field has a characteristic that is a prime number. This means that if you investigate a finite field—let's say it's defined by certain addition and multiplication operations—you will find that the number you get when you add the identity element (1) to itself repeatedly until you reach 0 is prime. It won't be composite (which would have factors).

Think of a game where a player can only collect prime-numbered treasure chests. Each time they reach a treasure chest (1), they can collect multiples (adding to themselves). However, if they ever reach a composite number of chests, they realize they can't open all the locked treasure chests without an extra step—just like how composite numbers cannot entirely fit the field characteristic.

So the proof will be by contradiction. So the theorem says that the characteristic should be prime number but as per the proof by contradiction strategy, I will assume the contrary and I will assume that the characteristic is not a prime number.

In a proof by contradiction, you start by assuming the opposite of what you want to prove is true. Here, instead of proving that the characteristic is prime, we assume it could be composite. From that assumption, we then deduce other statements that logically lead us to a conflict in our assumption, showing that the assumption was incorrect.

Imagine a detective trying to solve a mystery. They start by assuming that every clue points to the wrong suspect. However, as they investigate further, each clue leads them back to the same person, making them realize that their original assumption was incorrect. Similarly, assuming the field characteristic is composite leads to contradictions in our findings.

Now if this is the case then I am going to prove that either the element 1 added m times will give you the element 0 or the element 1 added to itself m times will give you the element 0.

By assuming that the characteristic is composite, we find m as the product of two numbers. The next step reveals that this assumption implies that if we add '1' to itself m times, we'll have two outcomes, either must yield zero, indicating that the characteristic must instead be less than m and prime.

Consider a train that stops at every 5th station. If you assume that the station count can also be counted in odd numbers and even numbers, each time you reach a station (1), you realize that either you have not technically counted those stations correctly or you've circled back around, indicating some stations (0) are unvisited, which prompts an exploration of the underlying structure.

So I get a contradiction and that shows that my claim is correct and since my claim is correct, then that contradicts the assumption that I made here namely the characteristic is a composite number.

Having established that reaching the assumption of a composite characteristic leads to contradictions—specifically that the operation can't consistently yield a satisfying outcome without looping back to a true prime characteristic—we conclude that the characteristic must be indeed a prime number.

Imagine you have a locked box (the field). You attempt to open it with a combination (the characteristic). Initially, you think the combination may include many numbers (composite). Each time you try, the lock simply won't twist. It's not until you use a straightforward prime number combination that the box finally opens, confirming that only prime numbers operate effectively in this cycle.

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