Glossary of field theory
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Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.
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Definition of a field
A field is an commutative ring (F,+,*) of which every nonzero element is invertible. Over a field, we can perform addition, subtraction, multiplication and division.
The abelian group of non-zero elements of a field F is typically denoted by F×;
- Characteristic
- The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Zp has characteristic p.
The ring of polynomials with coefficients in F is denoted by F[x].
Basic definitions
- Subfield
- A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
- Prime field
- A prime field is the unique smallest subfield of F.
- Extension field
- If F is a subfield of E then E is an extension field of F.
- Algebraic extension
- If an element α of an extension field E over F is the root of a polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E is an algebraic extension of F.
- Splitting field
- A field extension generated by the complete factorisation of a polynomial.
- Normal extension
- A field extension generated by the complete factorisation of a set of polynomials.
- Separable extension
- An extension generated by roots of separable polynomials.
- Primitive element
- A element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α.
- Perfect field
- A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.
- Algebraically closed field
- A maximal algebraic extension field of F is its algebraic closure. A field is algebraically closed if it is its own algebraic closure.
- Transcendental
- If an element is not algebraic over F, then it is transcendental.
- Transcendence degree
- The number of independent transcendental elements in a field extension. It is used to define the dimension of an algebraic variety.
Homomorphisms
- Field homomorphism
- A field homomorphism between two fields E and F is a function
- f : E → F
- such that
- f(x + y) = f(x) + f(y)
- and
- f(xy) = f(x) f(y)
- for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism
- f : E → F.
- The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, for example, complex conjugation.
Types of fields
- Finite field
- A field of finitely many elements.
- Ordered field
- A field with a total order compatible with its operations.
- Number field
- Algebraic extension of the field of rational numbers.
- Algebraic numbers
- The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory.
- Quadratic field
- A degree-two extension of the rational numbers.
- Cyclotomic field
- An extension of the rational numbers generated by a root of unity.
- Totally real field
- A number field generated by a root of a polynomial, having all its roots real numbers.
Galois theory
- Galois extension
- A normal, separable field extension.
- Galois group
- The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
- Kummer theory
- The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.
- Artin-Scheier theory
- Covers an exceptional case of Kummer theory, in characteristic p.
- Normal basis
- A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.
- Tensor product of fields
- A different foundational piece of algebra, including the compositum operation (join of fields).
Extensions of Galois theory
- Inverse problem of Galois theory
- Given a group G, find an extension of the rational number or other field with G as Galois group.
- Differential Galois theory
- The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie groups. It has not, probably, reached definitive form.
- Grothendieck's Galois theory
- A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.