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# algebraically closed

A field is algebraically closed if every polynomial with coefficients in the field has a root in the field.

Neither the field of rational numbers nor the field of real numbers is algebraically closed. For instance, the polynomial $$p(x)=x^2+2$$ does not have a either a real or a rational root. However, it does have complex roots ($$\pm i \sqrt{2}\,$$), and indeed the field of complex numbers is algebraically closed.

Citation Info

• [MLA] “algebraically closed.” Platonic Realms Interactive Mathematics Encyclopedia. Platonic Realms, 21 Feb 2013. Web. 21 Feb 2013. <http://platonicrealms.com/>
• [APA] algebraically closed (21 Feb 2013). Retrieved 21 Feb 2013 from the Platonic Realms Interactive Mathematics Encyclopedia: http://platonicrealms.com/encyclopedia/algebraically-closed/