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”, in 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 $\mathbf {Q} _{p}^{\mathrm {a} }$ is not complete, so we shall consider its completion $\mathbf {C} _{p}$: 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 $\mathbb {C}$ contains an algebraic closure of $\mathbb {Q}$, it is by no means the only algebraically closed field containing an algebraic closure of $\mathbb {Q}$. We denote by $\mathbb {Q} ^{\mathrm {alg} }$ the algebraic closure of $\mathbb {Q}$ in $\mathbb {C}$; this field is simply the subfield of $\mathbb {C}$ consisting of algebraic numbers. The field $\mathbb {Q} ^{\mathrm {alg} }$ is isomorphic, then, to any algebraic closure of $\mathbb {Q}$, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with $\mathbb {Q} ^{\mathrm {alg} }$ than we were.

Usage notes

• Notations for the algebraic closure of a field $F$ include ${\overline {F}}$ and $F^{\mathrm {a} }$.
• 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, $\mathbb {C}$, is the algebraic closure of the field of real numbers, $\mathbb {R}$.
• The algebraic closure of the field of p-adic numbers, $\mathbb {Q} _{p}$, is denoted ${\overline {\mathbb {Q} }}_{p}$ or $\mathbb {Q} _{p}^{\mathrm {a} }$. (Unlike $\mathbb {C}$, and indeed unlike $\mathbb {Q} _{p}$, ${\overline {\mathbb {Q} }}_{p}$ is not metrically complete: its metric completion, which is algebraically closed, is denoted $\mathbb {C} _{p}$ or $\Omega _{p}$.)

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^[1], archived from the original on 23 October 2014

Further reading

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