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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a field K is pseudo algebraically closed if it satisfies certain properties which hold for any algebraically closed field. In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F. As an example, the field of real numbers is not algebraically closed, because the polynomial equation x2 + 1 = 0 has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. Also, no finite field F is algebraically closed, because if a1, a2, , an are the elements of F, then the polynomial (x a1)(x a2) ··· (x an) + 1 has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.