Algebra over a field refers to the branch of mathematics that deals with the study of algebraic structures over a field. A field is a mathematical structure where arithmetic operations such as addition, subtraction, multiplication, and division are well-defined. Synonyms for algebra over a field include field theory, algebraic number theory, and Galois theory. These synonyms refer to various aspects of algebraic structures, including the properties of field extensions, the study of algebraic equations with coefficients in a field, and the symmetry groups of algebraic equations. Algebra over a field is an essential tool in many areas of mathematics and science, including cryptography, coding theory, and physics.