Field

Algebraically closed field

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

1. every non-constant polynomial has a root, i.e. a solution to ;
2. is an ^irreducible iff it is linear, i.e. ;
3. there does not exist a proper algebraic extension of ;
4. every maximal ideal of is of the form for some .

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 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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