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. <>
  • [APA] algebraically closed (21 Feb 2013). Retrieved 21 Feb 2013 from the Platonic Realms Interactive Mathematics Encyclopedia:


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