no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. By contrast, the fundamental...

algebraically closed field is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field...

equivalence of categories between smooth proper algebraic curves over an algebraically closed field F and finite field extensions of F(T). Historically, division...

a = −b2. F is not algebraically closed, but its algebraic closure is a finite extension. F is not algebraically closed but the field extension F ( − 1...

of K. The algebraic closure of K is also the smallest algebraically closed field containing K, because if M is any algebraically closed field containing...

mathematics, a field K {\displaystyle K} is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept...

Dimension of an algebraic variety for details. A product of finitely many algebraic varieties (over an algebraically closed field) is an algebraic variety. A...

that we are working over a fixed, algebraically closed field k (in classical algebraic geometry, k is usually the field of complex numbers). First, we define...

k. The separable closure of k is algebraically closed. Every reduced commutative k-algebra A is a separable algebra; i.e., A ⊗ k F {\displaystyle A\otimes...

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the...