Algebraically closed field
A field in which every nonconstant polynomial has a root; equivalently, every polynomial over it factors into linear factors.
Field (mathematics): A set with addition, subtraction, multiplication, and division by nonzero elements satisfying familiar arithmetic laws. Algebraic closure is a property of fields, defined through polynomials with coefficients in them.
Fundamental theorem of algebra: The theorem that every nonconstant complex polynomial has a complex root. It establishes that the complex numbers satisfy the defining root condition.
Complex number: A number of the form a + bi, where a and b are real and i² = −1. The complex numbers are the standard example of an algebraically closed field.
Hilbert's Nullstellensatz: A theorem relating polynomial equations over an algebraically closed field to ideals in polynomial rings. Its correspondence between ideals and zero sets depends on the coefficient field being algebraically closed.
Real number: An element of the complete ordered field that models the number line. The real field is not algebraically closed, since polynomials such as x² + 1 have no real root.
Polynomial: An expression formed from coefficients, variables, and nonnegative integer powers, combined by addition and multiplication. The closure condition quantifies over every nonconstant polynomial over the field.
Irreducible polynomial: A nonconstant polynomial that cannot be factored into polynomials of smaller positive degree over its coefficient field. Over an algebraically closed field, every irreducible polynomial has degree one.
Algebraic numbers: The complex numbers that are roots of nonzero polynomials with rational coefficients. Together they form an algebraically closed field, despite being countable.
Affine algebraic variety: A geometric set defined as the common zeros of polynomials over a field. Over an algebraically closed field, polynomial equations have the standard geometric interpretation through their points.
Perfect field: A field over which every algebraic extension is separable. Perfectness concerns repeated roots in extensions, not whether every polynomial already has a root.