algebraically closed

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Algebraically closed field

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algebraically closed (not comparable)

1. (algebra, field theory, of a field) Which contains as an element every root of every nonconstant univariate polynomial definable over it (i.e., over said field).

   The fundamental theorem of algebra states that the field of complex numbers, ${\displaystyle \mathbb {C} }$, is algebraically closed.

   • 2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 166:

     Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1. ${\displaystyle \square }$
     For instance, the fundamental theorem of algebra (Theorem III.8.11) states that ${\displaystyle \mathbb {C} }$ is algebraically closed. The fields ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {Q} }$, ${\displaystyle \mathbb {Z} _{p}}$, are not algebraically closed, but ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {Q} }$ can be embedded into the algebraically closed field ${\displaystyle \mathbb {C} }$.

   • 2008, M. Ram Murty, Problems in Analytic Number Theory, 2nd edition, Springer, page 155:

     In many ways ${\displaystyle \mathbb {Q} _{p}}$ is analogous to ${\displaystyle \mathbb {R} }$. For example, ${\displaystyle \mathbb {R} }$ is not algebraically closed. The exercises below show that ${\displaystyle \mathbb {Q} _{p}}$ is not algebraically closed. However, by adjoining ${\displaystyle i={\sqrt {-1}}}$ to ${\displaystyle \mathbb {R} }$, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure ${\displaystyle {\overline {\mathbb {Q} }}_{p}}$ of ${\displaystyle \mathbb {Q} _{p}}$ is not of finite degree over ${\displaystyle \mathbb {Q} }$. Moreover, ${\displaystyle \mathbb {C} }$ is complete with respect to the extension of the usual norm of ${\displaystyle \mathbb {R} }$. Unfortunately, ${\displaystyle {\overline {\mathbb {Q} }}_{p}}$ is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by ${\displaystyle \mathbb {C} _{p}}$, and it turns out to be both algebraically closed and complete.

   • 2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 195:

     Definition 5.3.14. Let ${\displaystyle F}$ be a field. A field extension ${\displaystyle P}$ is called an algebraic closure of ${\displaystyle F}$ if ${\displaystyle P}$ is algebraically closed and every proper subfield of ${\displaystyle P}$ containing ${\displaystyle F}$ is not algebraically closed.
     In other words, the algebraic closure of ${\displaystyle F}$ is the minimal algebraically closed field containing ${\displaystyle F}$.

2. (algebra, group theory, of a group) Such that any finite set of equations and inequations has a solution.

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