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If F and G are fields and G contains F, then the field extension G/F is called algebraic if every element of G is algebraic over F, meaning that for every element x of G there exists a non-zero polynomial p with coefficients in F such that p(x) = 0. |
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If F and G are fields and G contains F, then the field extension G/F is called algebraic if every element of G is algebraic over F, meaning that for every element x of G there exists a non-zero polynomial p with coefficients in F such that p(x) = 0. |
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F[x] is a field, and it is unique up to isomorphism if and only if |
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F[x], the set of all polynomials in x with coefficients in F, is a field, and it is unique up to isomorphism if and only if |
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(Notation: F[x] is the ring of polynomials over F, that is, their coefficents, arguments, and values are elements of F.) |
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