It is not a philosophical subjet but the real solution of this division.
How something so simple not be seen before ?
I don't understand.
But you just need to know Galois fields and  the "Théorie des ensembles".  After read well what I go to say and you go to discover this just looking well.
You need to have eyes to see. 
So simple and so marvelous solution when you start to understand 
 
