Algebraically Complete Fields

Abstract Algebra · Axiom Academy

INTRO Algebraically Closed Fields Why can't we solve every polynomial equation in ℝ? Discover the power of algebraic closure! Let's try to solve x^2 + 1 = 0 using real numbers. Try different values of x and see what happens! When we graph y = x^2 + 1 , we can see why there are no real roots. Try different polynomials: Let's introduce a new number: i , where i^2 = -1 . Now we can solve x^2 + 1 = 0 ! The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one complex root. This means a polynomial of degree n has exactly n roots (counting multiplicity) in . A field F is algebraically closed if every non-constant polynomial with coefficients in F has at least one root in F . Equivalently, every polynomial of degree n over F splits into n linear factors over F . Factorization: In an algebraically closed field, every polynomial factors completely into linear terms Simplicity: We don't need to keep extending our field to solve equations Theory: Many theorems in algebra and analysis rely on algebraic closure Geometry: Algebraically closed fields have nice geometric properties (e.g., no "missing" intersection points)

This is the written version of the interactive lesson above. See the full Abstract Algebra course.