Algebraically closed field

A field $𝐾$ is called algebraically closed iff it satisfies the following equivalent properties #m/def/ring

every non-constant polynomial $𝑝 (𝑥) \in 𝐾 [𝑥]$ has a root, i.e. a solution to $𝑝 (𝑥) = 0$ ;
$𝑝 (𝑥) \in 𝐾 [𝑥]$ is an irreducible polynomial iff it is linear, i.e. $d e g 𝑝 = 1$ ;
there does not exist a proper algebraic extension of $𝐾$ ;
every maximal ideal of $𝐾 [𝑥]$ is of the form $⟨ 𝑥 - 𝛼 ⟩$ for some $𝛼 \in 𝐾$ .

Assuming choice, every field is contained in an algebraically closed one: a/the Algebraic closure.

Examples and nonexamples

$ℝ$ is not algebraically closed, since $𝑥 2 + 1$ has no real root.
$ℂ$ is the closure of the real numbers.

Properties

Division algebra with only algebraic elements over an algebraically closed field

#state/tidy | #lang/en | #SemBr