Glossary of field theory

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field.
A field is a commutative ring ("F",+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field "F" form an abelian group under multiplication; this group is typically denoted by "F"×;
The ring of polynomials in the variable "x" with coefficients in "F" is denoted by "F"["x"].
Field extensions.
Let "E" / "F" be a field extension.