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

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

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

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...

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...

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

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...

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

.} An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E. An example is the field of complex...

Number field Finite extension of the field of rational numbers. Algebraic numbers The field of algebraic numbers is the smallest algebraically closed extension...