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Algebraic Closure

An algebraic closure of F is a field F^_ such that F^_ is algebraic over F and every polynomial f(x) in F[x] splits completely over F^_, so that F^_ can be said to contain all the elements that are algebraic over F.

For example, the field of complex numbers C is the algebraic closure of the field of reals R.

See also

Algebraically Closed, Splitting Field

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References

Dummit, D. S. and Foote, R. M. "Splitting Fields and Algebraic Closures." §13.4 in Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley, pp. 536-544, 2004.

Referenced on Wolfram|Alpha

Algebraic Closure

Cite this as:

Weisstein, Eric W. "Algebraic Closure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AlgebraicClosure.html

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