algebraically closed

Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1. #92;square For instance, the fundamental theorem of algebra (Theorem III.8.11) states that #92;Complex is algebraically closed. The fields #92;R, #92;Q, #92;Z#95;p, are not algebraically closed, but #92;R and #92;Q can be embedded into the algebraically closed field #92;Complex.

2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 166

In many ways #92;Q#95;p is analogous to #92;R. For example, #92;R is not algebraically closed. The exercises below show that #92;Q#95;p is not algebraically closed. However, by adjoining i#61;#92;sqrt#123;-1#125; to #92;R, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure #92;overline#92;Q#95;p of #92;Q#95;p is not of finite degree over #92;Q. Moreover, #92;Complex is complete with respect to the extension of the usual norm of #92;R. Unfortunately, #92;overline#92;Q#95;p is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by #92;Complex#95;p, and it turns out to be both algebraically closed and complete.

2008, M. Ram Murty, Problems in Analytic Number Theory, 2nd edition, Springer, page 155

Definition 5.3.14. Let F be a field. A field extension P is called an algebraic closure of F if P is algebraically closed and every proper subfield of P containing F is not algebraically closed. In other words, the algebraic closure of F is the minimal algebraically closed field containing F.

2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 195