Glossaryfield theory
Field theory isbranchmathematicswhich fieldssstudied. This isglossarysome terms ofsubject.
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2 Basic definitions 3 Homomorphisms 4 Typesfields |
Definition offield
A fieldan commutative ring (F,+,*)which every nonzero elementinvertible. Overfield, we can perform addition, subtraction, multiplicationdivision.
The abelian groupnon-zero elements offield Ftypically denoted by F×;
;Characteristic : The characteristic offield F issmallest positive integer n such that n·1 = 0; here n·1 standsn summands 1 + 1 + 1 + ... + 1. If no such n exists, we saycharacteristiczero. Every non-zero characteristic isprime number. For example,rational numbers,real numbers andp-adic numbers have characteristic 0, whilefinite field Zp has characteristic p.
The ringpolynomialscoefficientsFdenoted by F[x].
Basic definitions
; Subfield : A subfield offield F issubsetF whichclosed underfield operation +*Fwhich,these operations, forms itselffield.
; Prime field : A prime field isunique smallest subfieldF.
; Extension field : If F issubfieldE then Ean extension fieldF.
; Algebraic extension : If an element αan extension field E over F isroot ofpolynomialF[x], then αalgebraic over F. If every elementEalgebraic over F, then Ean algebraic extensionF.
; Primitive element : A element αan extension field E overfield Fcalledprimitive element if E=F(α),smallest extension field containing α.
; Algebraically closed field : The largest unique algebraic extension fieldF.
; Transcendental : If an elementnot algebraic over F, then ittranscendental.
Homomorphisms
; Field homomorphism : A field homomorphism between two fields EF isfunction f : E -> F such that f(x + y) = f(x) + f(y)f(xy) = f(x) f(y)all x, yE, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1xEx ≠ 0,that finjective. Fields, togetherthese homomorphisms, formcategory. Two fields EFcalled isomorphic if there existsbijective homomorphism f : E -> F. The two fieldsthen identicalall practical purposes.
Typesfields
; Finite field : A fieldfinitely many elements.
; Ordered field : A field withtotal order compatibleits operations.
; Number field : Algebraic extension offieldrational numbers.
; Algebraic numbers : The fieldalgebraic numbers isalgebraically closed extension offieldrational numbers.
