- 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. (Seefield theory (physics) for the unrelated field theories in physics.)**Definition of a field**A

**field**is acommutative 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"] .

**Basic definitions**;

**Characteristic**: The "characteristic" of the field "F" is the smallest positiveinteger "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 aprime number . For example, therational number s, thereal number s and the "p"-adic numbers have characteristic 0, while the finite field**Z**_{"p"}has characteristic "p".;

**Subfield**: A "subfield" of a field "F" is asubset of "F" which is closed under the field operation + and * of "F" and which, with these operations, forms itself a field.;

: The "prime field" of the field "F" is the unique smallest subfield of "F".Prime field ;

**Extension field**: If "F" is a subfield of "E" then "E" is an "extension field" of "F". We then also say that "E"/"F" is a "field extension".;

**Degree of an extension**: Given an extension "E"/"F", the field "E" can be considered as avector space over the field "F", and the dimension of this vector space is the "degree" of the extension, denoted by ["E" : "F"] .;

**Finite extension**: A "finite extension" is a field extension whose degree is finite.;

: If an element α of an extension field "E" over "F" is theAlgebraic extension root of a non-zero polynomial in "F" ["x"] , then α is "algebraic" over "F". If every element of "E" is algebraic over "F", then "E"/"F" is an "algebraic extension".;

**Generating set**: Given a field extension "E"/"F" and a subset "S" of "E", we write "F"("S") for the smallest subfield of "E" that contains both "F" and "S". It consists of all the elements of "E" that can be obtained by repeatedly using the operations +,-,*,/ on the elements of "F" and "S". If "E" = "F"("S") we say that "E" is generated by "S" over "F".;

**Primitive element**: An 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. Such an extension is called a.simple extension ;

: A field extension generated by the complete factorisation of a polynomial.Splitting field ;

: A field extension generated by the complete factorisation of a set of polynomials.Normal extension ;

: An extension generated by roots ofSeparable extension separable polynomial s.;

: A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.Perfect field ;

: A field "F" is "algebraically closed" if every polynomial in "F" ["x"] has a root in "F"; equivalently: every polynomial in "F" ["x"] is a product of linear factors.Algebraically closed field ;

: An "algebraic closure" of a field "F" is an algebraic extension of "F" which is algebraically closed. Every field has an algebraic closure, and it is essentially unique.Algebraic closure ;

**Transcendental**: Those elements of an extension field of "F" that are not algebraic over "F" are "transcendental" over "F".;

**Algebraically independent elements**: Elements of an extension field of "F" are "algebraically independent" over "F" if they don't satisfy any polynomial equation with coefficients in "F".;

: The number of algebraically independent transcendental elements in a field extension. It is used to define theTranscendence degree 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" isinjective . Fields, together with these homomorphisms, form a category. Two fields "E" and "F" are called**isomorphic**if there exists abijective 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**;

: A field with finitely many elements.Finite field ;

: A field with aOrdered field total order compatible with its operations.;

Rational number s;

Real number s;

Complex number s;

: Finite extension of the field of rational numbers.Number field ;

: The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied inAlgebraic number salgebraic number theory .;

: A degree-two extension of the rational numbers.Quadratic field ;

: An extension of the rational numbers generated by aCyclotomic field root of unity .;

: A number field generated by a root of a polynomial, having all its roots real numbers.Totally real field ;

Formally real field ;

Real closed field Galois theory ;

: A normal, separable field extension.Galois extension ;

: TheGalois group 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 areprofinite group s.;

: The Galois theory of taking "n"-th roots, given enoughKummer theory roots of unity . It includes the general theory ofquadratic extension s.;

: Covers an exceptional case of Kummer theory, in characteristic "p".Artin-Schreier theory ;

: A basis in the vector space sense of "L" over "K", on which the Galois group of "L" over "K" acts transitively.Normal basis ;

: A different foundational piece of algebra, including theTensor product of fields compositum operation (join of fields).**Extensions of Galois theory**;

: Given a group "G", find an extension of the rational number or other field with "G" as Galois group.Inverse problem of Galois theory ;

: The subject in which symmetry groups ofDifferential Galois theory differential equation s are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations whenSophus Lie founded the theory ofLie group s. It has not, probably, reached definitive form.;

Grothendieck's Galois theory : A very abstract approach fromalgebraic geometry , introduced to study the analogue of thefundamental group .

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