Algebraic Closure Definition

Abstract Algebra · Axiom Academy

Understanding when a field contains all the roots it could possibly need, and how to construct such completions. A field F is algebraically closed if every non-constant polynomial in F[x] has at least one root in F . 2. Equivalent Characterization Being algebraically closed is equivalent to a stronger property: every polynomial splits completely into linear factors. Why are these equivalent? If f has a root , then for some . If F is algebraically closed, g(x) also has a root, and we can factor again. By induction, f splits completely. 3. The Fundamental Theorem of Algebra This remarkable theorem tells us that every polynomial with complex coefficients has a complex root. In other words, needs no further extension to solve polynomial equations. Every field has an algebraic closure, and it's essentially unique.

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