algebraic closure

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 ${\displaystyle \mathbf {Q} _{p}^{\mathrm {a} }}$ 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.

• 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 ${\displaystyle \Omega _{p}}$.)

• algebraically closed

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

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