The Artin-Schreier Theorem characterizes fields whose algebraic closure is a finite, nontrivial extension of the field itself: any such field must be real closed, and the extension to its algebraic closure has degree exactly two. Named for Emil Artin and Otto Schreier, it explains why the real numbers, whose algebraic closure is the complex numbers, a degree-two extension, are essentially the only kind of field that can stand in that relationship to its own algebraic closure.
Facts
StatementIf F is an ordered field, then F has an algebraic extension called the real closure of F, unique up to a unique isomorphism fixing F, that is itself a real closed field whose ordering extends the given ordering on F. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Artin-Schreier Theorem (Wikipedia)
Wikimedia FoundationReal closure section, statement of the theorem
If F is an ordered field, the Artin-Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F
Real closure section, attribution and year
The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926.
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