algebraic closure

algebraic closure

English

Noun

algebraic closure (plural algebraic closures)

1. (algebra, field theory, of a field F) 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).
   • 1999, Shreeram S. Abhyankar, Galois Theory of Semilinear Transformations, Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller (editors), Aspects of Galois Theory, Cambridge University Press, page 1,

     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.

   • 2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127,

     It turns out that the algebraic closure ${\\displaystyle \\mathbf {Q} _{p}^{\\mathrm {a} }}$ is not complete, so we shall consider its completion ${\\displaystyle \\mathbf {C} _{p}}$: This field turns out to be algebraically closed and is a natural domain for the study of "analytic functions."

   • 2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179,

     While ${\\displaystyle \\mathbb {C} }$ contains an algebraic closure of ${\\displaystyle \\mathbb {Q} }$, it is by no means the only algebraically closed field containing an algebraic closure of ${\\displaystyle \\mathbb {Q} }$. We denote by ${\\displaystyle \\mathbb {Q} ^{\\mathrm {alg} }}$ the algebraic closure of ${\\displaystyle \\mathbb {Q} }$ in ${\\displaystyle \\mathbb {C} }$; this field is simply the subfield of ${\\displaystyle \\mathbb {C} }$ consisting of algebraic numbers. The field ${\\displaystyle \\mathbb {Q} ^{\\mathrm {alg} }}$ is isomorphic, then, to any algebraic closure of ${\\displaystyle \\mathbb {Q} }$, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with ${\\displaystyle \\mathbb {Q} ^{\\mathrm {alg} }}$ than we were.

Usage notes

• Notations for the algebraic closure of a field ${\\displaystyle F}$ include ${\\displaystyle {\\overline {F}}}$ and ${\\displaystyle F^{\\mathrm {a} }}$.
• Using Zorn's lemma (or the weaker ultrafilter lemma), it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Consequently, authors often speak of the (rather than an) algebraic closure of K. (See Algebraic closure on Wikipedia.Wikipedia )
• The field of complex numbers, ${\\displaystyle \\mathbb {C} }$, is the algebraic closure of the field of real numbers, ${\\displaystyle \\mathbb {R} }$.
• The algebraic closure of the field of p-adic numbers, ${\\displaystyle \\mathbb {Q} _{p}}$, is denoted ${\\displaystyle {\\overline {\\mathbb {Q} }}_{p}}$ or ${\\displaystyle \\mathbb {Q} _{p}^{\\mathrm {a} }}$. (Unlike ${\\displaystyle \\mathbb {C} }$, and indeed unlike ${\\displaystyle \\mathbb {Q} _{p}}$, ${\\displaystyle {\\overline {\\mathbb {Q} }}_{p}}$ is not metrically complete: its metric completion, which is algebraically closed, is denoted ${\\displaystyle \\mathbb {C} _{p}}$ or ${\\displaystyle \\Omega _{p}}$.)

Related terms

• algebraically closed

Translations

field which is algebraically closed over another one

Finnish: algebrallinen sulkeuma Portuguese: fecho algébrico m

References

• Frédérique Oggier (2010), “Introduction to Algebraic Number Theory”, in ntu.edu.sg/~frederique/Teaching‎, archived from the original on 2014-10-23, retrieved 2012-09-21

Further reading

• Closure (mathematics) on Wikipedia.Wikipedia
• Algebraically closed field on Wikipedia.Wikipedia
• Algebraic closure on Encyclopedia of Mathematics
• Algebraic Closure on Wolfram MathWorld