The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings.
One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups,
rings, invariant theory, and
cohomology. This is the meaning mathematicians associate
with the word "algebra." When there is the possibility of confusion, this
field of mathematics is often referred to as abstract
algebra.
The word "algebra" can also refer to the "school algebra" generally taught in American middle and high schools. This includes the solution of polynomial
equations in one or more variables and basic properties of functions
and graphs. Mathematicians call this subject "elementary
algebra," "high school algebra," "junior high school algebra,"
or simply "school algebra," reserving the word "algebra" for
the more advanced aspects of the subject.
Finally, the word is used in a third way, not as a subject area but as a particular type of algebraic structure. Formally, an algebra is a vector
space
over a field with a multiplication. The multiplication must be distributive
and, for every
and must satisfy
Examples of algebras include the algebra of real numbers, vectors and matrices, tensors, complex numbers,
and quaternions. (Note that linear
algebra, which is the study of linear sets of equations and their transformation
properties, is not an algebra in the formal sense of the word.) Other more exotic
algebras that have been investigated and found to be of interest are usually named
after one or more of their investigators. This practice unfortunately leads to entirely
unenlightening names which are commonly used by algebraists without further explanation
or elaboration.
over a field 
with a multiplication. The multiplication must be distributive
and, for every 
and 
must satisfy
