Field

Algebraically closed field

A field 𝐾 is called algebraically closed iff it satisfies the following equivalent properties ring

1. every non-constant polynomial 𝑝(𝑥) ∈𝐾[𝑥] has a root, i.e. a solution to 𝑝(𝑥) =0;
2. 𝑝(𝑥) ∈𝐾[𝑥] is an ^irreducible iff it is linear, i.e. deg⁡𝑝 =1;
3. there does not exist a proper algebraic extension of 𝐾;
4. every maximal ideal of 𝐾[𝑥] is of the form ⟨𝑥 −𝛼⟩ for some 𝛼 ∈𝐾.

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

Examples and nonexamples

• Real numbers is not algebraically closed, since 𝑥2 +1 has no real root.
• Complex numbers is the closure of the real numbers.

Properties

• Division algebra with only algebraic elements over an algebraically closed field

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