1. the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. = abstract algebra
2. a particular type of algebraic structure. Formally, an algebra is a vector space V over a field F with a multiplication, which is distributive, and for every f from F and every x,y from V
Etymology:
from the title of a work written around 825 by the Arabic mathematician known as al-Khowarizmi, entitled al-jebr w' al-muqabalah. In Arabic, al- is the definite article "the." The first noun in the title is jebr"reunion of broken parts," from the verb jabara"to reunite, to consolidate." The second noun is from the verb qabala, with meanings that include "to place in front of, to balance, to oppose, to set equal." Together the two nouns describe some of the manipulations so common in algebra: combining like terms, transposing a term to the opposite side of an equation, setting two quantities equal, etc.
[Schwartzman, S. (2012). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms used in English. Washington: The Mathematical Association of America.]
