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Field


A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. An archaic name for a field is rational domain. The French term for a field is corps and the German word is Körper, both meaning "body." A field with a finite number of members is known as a finite field or Galois field.

Because the identity condition is generally required to be different for addition and multiplication, every field must have at least two elements. Examples include the complex numbers (C), rational numbers (Q), and real numbers (R), but not the integers (Z), which form only a ring.

It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of complex numbers.


See also

Adjunction, Base Field, Coefficient Field, Cyclotomic Field, Division Algebra, Extension Field, Field Axioms, Field Characteristic, Finite Field, Function Field, Local Field, Mac Lane's Theorem, Module, Number Field, Pythagorean Field, Quadratic Field, Ring, Splitting Field, Subfield, Vector Field Explore this topic in the MathWorld classroom

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References

Allenby, R. B. Rings, Fields, and Groups: An Introduction to Abstract Algebra, 2nd ed. Oxford, England: Oxford University Press, 1991.Dummit, D. S. and Foote, R. M. "Field Theory." Ch. 13 in Abstract Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, pp. 422-470, 1998.Ellis, G. Rings and Fields. Oxford, England: Oxford University Press, 1993.Ferreirós, J. "A New Fundamental Notion for Algebra: Fields." §3.2 in Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Basel, Switzerland: Birkhäuser, pp. 90-94, 1999.Joye, M. "Introduction élémentaire à la théorie des courbes elliptiques." http://www.dice.ucl.ac.be/crypto/introductory/courbes_elliptiques.html.Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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Field

Cite this as:

Weisstein, Eric W. "Field." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Field.html

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