Word.ws English Dictionary

algebraic closure

algebraic closure noun: a field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

Noun

algebraic closure (plural algebraic closures)

  1. (field theory) A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).
    • The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University...
    • It turns out that the algebraic closure #92;mathbf#123;Q#125;#92;mathrm#123;a#125;#95;p is not complete, so we shall consider its completion #92;mathbf#123;C#125;#95;p: This field turns out to be algebraically closed and is a natural domain for the study of "analytic functions."2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127

Forms of algebraic closure

pluralalgebraic closures

Questions about algebraic closure

What does algebraic closure mean?
Algebraic closure means a field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).
What part of speech is algebraic closure?
Algebraic closure is a noun.
What is the plural of algebraic closure?
The plural of algebraic closure is algebraic closures.

Words near algebraic closure in the dictionary

  1. algebra
  2. algebra over a field
  3. algebra over a ring
  4. algebraic
  5. algebraic analysis
  6. algebraic closure
  7. algebraic closure operator
  8. algebraic closures
  9. algebraic combinatorics
  10. algebraic curve
  11. algebraic curves

Source and license

Adapted from Wiktionary: algebraic closure (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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