What is an equation? Few people know that. What is an algebraic
equation? When is such an equation or system of equations solvable?
For instance, xyz=0 and zyx=8 is clearly unsolvable in the class of
commutative rings with identity, but there exists a solution in the
set of 2x2-matrices over the reals. We might fix an algebraic
structure A and look if a system of equations has a solution in A or
in an extension B of A.
We get a criterion for solvability in some extension by a
generalization of Hilbert́s Nullstellensatz. But very strange things
can happen. For instance, an equation can be solvable in an extension
C of A, but not in an extension of B (as above). We also touch the
theory of algebraically closed groups and other stuctures, and also
algorithmic aspects like Groebner bases.