Branches of Mathematics
Field Theory
Citation Formats
General Reference
APA Style
BibTeX
Field theory is the branch of abstract algebra that studies fields, sets on which addition, subtraction, multiplication and division are defined and behave as they do on the rational numbers, together with the extensions built by adjoining new elements to a smaller field.
Facts
Central QuestionWhich polynomial equations can be solved by radicals, and what structure, made precise by Galois theory's correspondence between field extensions and permutation groups, governs when such a solution exists. 1 Key DebateWhether a general algebraic equation of degree five or higher could be solved by radicals at all. Evariste Galois settled the question by characterizing solvability in terms of the permutation group of an equation's roots, a correspondence between field extensions and group theory that only reached wide acceptance among mathematicians after his death. 1 Cross-Tradition Connections
Sources
1. Field (mathematics) (Wikipedia)
Wikimedia FoundationLead sectionQuote, Lead section
A field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
1. Field (mathematics) (Wikipedia)
Wikimedia FoundationGalois theory sectionQuote, Galois theory section
if the Galois group of a Galois extension as above is not solvable (cannot be built from abelian groups), then the zeros of f cannot be expressed in terms of addition, multiplication, and radicals
1. Field (mathematics) (Wikipedia)
Wikimedia FoundationHistory sectionQuote, History section
In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Korper, which means 'body' or 'corpus'
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
In mathematics, Galois theory, originally introduced by Evariste Galois, provides a connection between field theory and group theory.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaApplication to classical problems sectionQuote, Application to classical problems section
Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots.
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.
View At A Past Year
The atlas records no dated fact of its own for this entry, so there is no other year to choose.