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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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 Classification
Pure or Applied Connections
Associated With
Includes
Source Ernst Steinitz, Biography (MacTutor History of Mathematics)
Source Wolfram MathWorld: Jacobian Conjecture
Source Wolfram MathWorld: Mordell-Weil Theorem
Source Primitive element theorem (Wikipedia)
Source Radical Extension (Wikipedia)
Sources
1. Field (mathematics) (Wikipedia)
Wikimedia FoundationLead 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.
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
History
In 1871 Richard Dedekind introduced, for a set of real or complex numbers that is closed under the four arithmetic operations, the German word Körper, which means "body" or "corpus" (to suggest an organically closed entity).
View the Source Radical Extension (Wikipedia)
Includes: Radical Extension, Lead sentenceQuote, Includes: Radical Extension, Lead sentence
In mathematics and more specifically in field theory, a radical extension of a field K is a field extension obtained by a tower of
Primitive element theorem (Wikipedia)
Includes: Primitive Element Theorem, Lead sentenceQuote, Includes: Primitive Element Theorem, Lead sentence
In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e.
Wolfram MathWorld: Jacobian Conjecture
Wolfram MathWorldIncludes: Jacobian Conjecture, Subject classificationsQuote, Includes: Jacobian Conjecture, Subject classifications
Algebra Field Theory
Wolfram MathWorld: Mordell-Weil Theorem
Wolfram MathWorldIncludes: Mordell-Weil Theorem, Subject classificationsQuote, Includes: Mordell-Weil Theorem, Subject classifications
Algebra Field Theory
Ernst Steinitz, Biography (MacTutor History of Mathematics)
Includes: Ernst Steinitz, SummaryQuote, Includes: Ernst Steinitz, Summary
Ernst Steinitz worked on the theory of fields.
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