algebraically closed field

Definition

A field is called algebraically closed if any of the following conditions hold:

1. All non-constant polynomials over have a root in .

2. An irreducible polynomial over has degree 1 (Note by division algorithm)

3. All polynomials over split over (are products of degree 1 polynomials over )

Example

By the fundamental theorem of algebra, is algebraically closed.

Relation to algebraic field extensions

Let be a field. The following are equivalent:

1. is algebraically closed.

2. If is algebraic, then .

3. If is an extension and is algebraic over then

Proof

: Assume is algebraic. Let , then so that means there exists a polynomial such that . is algebraically closed, so

Therefore, .

: Trivial

: Let be an irreducible polynomial. Then is an algebraic extension (dimension over k is bounded by the degree of ). By assumption, then each element in so

which implies that . By condition 2 in the definition, is algebraically closed.

Relation to maximal ideals

Let be a field. Then is algebraically closed if and only if any maximal ideal of is of the form for a unique .

Proof

Assume is algebraically closed. Let be a maximal ideal. Since is a field, then is a PID, so for some . Assume for contradiction that . Then since is algebraically closed,

Therefore,

so would not be maximal, a contradiction. Thus, .

Assume conversely that every maximal ideal is of the form . Let be non-constant. Then for some maximal ideal (Zorn’s lemma). This implies that for some polynomial , and . Thus is algebraically closed.

Relation to maximal spectrum.

This means that the maximal spectrum of is bijective with . In other words, there aren’t any “other point” in that aren’t present in .