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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[edit]

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].

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Springer-Verlag