Usually it is said that linear equation systems have no solution, exactly one solution, or infinitely many solutions. But what if the equation is asked in the context of a finite field $K$? For example the equation
$$0x=0$$ Would be fulfilled by every element of the field, which would mean there is a finite number of solutions, but more than one. Is that correct?
Considering that the number of elements in a finite field is a power of a prime, would it be possible to construct a linear equation system with an exact number of solutions which is not a power of a prime?
The set of solutions will be a subspace of the vector space over the field (or a translate of a subspace) so it will have prime power cardinality.