Knowra Algebraic closure Algebraic closure An algebraically closed field is a field in which every nonconstant polynomial has a root. An algebraic closure of a field is an algebraic extension that is algebraically closed and minimal over that field.
Field (mathematics) : A field is a set with addition, subtraction, multiplication, and division by nonzero elements, satisfying familiar arithmetic laws. Algebraic closure begins with a field whose polynomials and extensions are being considered.
Fundamental theorem of algebra : Every nonconstant polynomial with complex coefficients has a complex root, and therefore factors completely over the complex numbers. It establishes directly that the complex numbers are algebraically closed.
Galois theory : Galois theory relates field extensions to groups of automorphisms that preserve a base field. Automorphisms of an algebraic closure encode the Galois groups used to study polynomial equations.
Real numbers : The real numbers form a complete ordered field containing the rational numbers. The real field is not algebraically closed because some real polynomials have no real roots.
Polynomial : A polynomial is an expression formed from coefficients, variables, and nonnegative integer powers combined by addition and multiplication. The defining condition asks whether every nonconstant polynomial over the field has a root.
Zorn's lemma : Zorn's lemma states that a partially ordered set in which every chain has an upper bound contains a maximal element. It supports existence proofs for maximal algebraic extensions, from which algebraic closures can be obtained.
Absolute Galois group : The absolute Galois group of a field is the group of automorphisms of its separable closure fixing that field. It captures the symmetries of algebraic roots across all finite extensions.
Complex numbers : The complex numbers are numbers of the form a + bi, where a and b are real and i² = −1. The complex field is algebraically closed, making it a familiar example of an algebraic closure of the reals.
Root of a polynomial : A root of a polynomial is an input at which the polynomial evaluates to zero. Algebraic closure guarantees such a root for every nonconstant polynomial.
Maximal algebraic extension : A maximal algebraic extension is an algebraic field extension that admits no proper algebraic extension within a chosen ambient field. Maximality forces the extension to contain roots for every polynomial over it.
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