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Algebraically closed field - Wikipedia

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In mathematics, a field F is algebraically closed if every non-constant polynomial in F[x] (the univariate polynomial ring 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 𝑥 2 + 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. 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. 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. However, the union of all finite fields of a fixed characteristic p is an algebraically closed field, w

Algebraically closed field - Wikipedia Jump to content From Wikipedia, the free encyclopedia Algebraic structure where all polynomials have roots This article includes a list of references , related reading , or external links , but its sources remain unclear because it lacks inline citations . Please help improve this article by introducing more precise citations. ( September 2021 ) ( Learn how and when to remove this message ) In mathematics , a field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F . In other words, a field is algebraically c

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