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.
- Algebraic extension
- An extension in which every element of E is algebraic over F.
- Simple extension
- An extension which is generated by a single element, called a primitive element, or generating element. The primitive element theorem classifies such extensions.
- Normal extension
- An extension that splits a family of polynomials: every root of the minimal polynomial of an element of E over F is also in E.
- Separable extension
- An algebraic extension in which the minimal polynomial of every element of E over F is a separable polynomial, that is, has distinct roots.
- Galois extension
- A normal, separable field extension.
- Primary extension
- An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equivalently, E is linearly disjoint from the separable closure of F.
- Purely transcendental extension
- An extension E/F in which every element of E not in F is transcendental over F.
- Regular extension
- An extension E/F such that E is separable over F and F is algebraically closed in E.
- Simple radical extension
- A simple extension E/F generated by a single element α satisfying αn = b for an element b of F. In characteristic p, we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension.
- Radical extension
- A tower F = F0 < F1 < ⋅⋅⋅ < Fk = E where each extension Fi / Fi−1 is a simple radical extension.
- Self-regular extension
- An extension E/F such that E ⊗F E is an integral domain.
- Totally transcendental extension
- An extension E/F such that F is algebraically closed in F.
- Distinguished class
- A class C of field extensions with the three properties
- If E is a C-extension of F and F is a C-extension of K then E is a C-extension of K.
- If E and F are C-extensions of K in a common overfield M, then the compositum EF is a C-extension of K.
- If E is a C-extension of F and E > K > F then E is a C-extension of K.