Today I shall discuss a problem from the National Maths Olympiad from 2000, the year I was born. Judging by the usual schedule of the Olympiad, this problem is probably my twin. It caught my eye because it literally disproves Fermat's Last Theorem on finite rings that are not fields (the title already said that). The FLT is, of course, a much harder problem than what I'm about to solve in the next few lines, but that doesn't make this problem less important.
Let (A,+,*) be a finite ring with 1 different from 0. Prove that the following affirmations are equivalent:
1. A is not a field.
2. The equation \(x^n+y^n=z^n\) has a solution for every natural n.
Solution:
We shall first prove that 2 implies 1. Let us presume by absurd that A is a field. Thus, since A is a field, (A\{0},*) is a finite group, with an order equal to a certain \(t\). For that \(t\), \(x^t=1\), for all elements of A\{0}. Thus, since the equation \(x^t+y^t=z^t\) has a solution, \(1+1=1\), so \(1=0\), which is false. Thus, A is not a field.
Conversely, if A is not a field, it means that it has a non-invertible element, let's call it \(x\). Since A is a finite ring, there are \(i<j\) such that \(x^i=x^j\), and, thus, \(x^i (x^{j-i}-1)=0\). By writing \(t=x^{i-1}(x^{j-1}-1)\), we have that \(x*t=t*x=0\). Now, let us take \((x+t)^n\). By the binomial theorem, this is a sum of combinations which revolve around \(x\) and \(t\), and, because \(x*t=t*x=0\), \((x+t)^n=x^n+t^n\), for all n, and, thus, we have a solution of the equation for every n.
Don't you think that this problem is really cute? I mean, it came 5 years after the proof of the FLT, and I am certain that the guy who proposed the problem had this in mind. I hope that my proof is correct, judging by the fact that the FLT has the unique record of having received the greatest amount of wrong solutions. Apparently, the peeps at the Wolfskehl committee received about 3 meters (in height) of wrong proofs. That might not sound like much, but when you think about the amount of pages required, you'll figure out that it's a really big lot. Tune in next time for trigonometric functions defined on complex numbers, another abstract algebra problem, or something completely random!
Cris.
