It relates algebraic sets to ideals in polynomial rings over algebraically closed fields. Moreover, it shows that quintic equations are algebraically unsolvable. This can be rephrased by saying that the field of algebraic numbers is algebraically closed. The dot product may be defined algebraically or geometrically. Over an algebraically closed field, diagonalizable matrices are equivalent to semi-simple matrices. PageRank can be computed either iteratively or algebraically. Let formula 74 be an irreducible representation of formula 71 over an algebraically closed field. A finite field is not algebraically closed. Let be an algebraically closed field and let be the projective "n"-space over . Singular values are similar in that they can be described algebraically or from variational principles. The theory of algebraically closed fields has quantifier elimination. Analytic spaces over algebraically closed fields are coherent. Because of this fact, theorems that hold "for any algebraically closed field", apply to C. Let now be a connected reductive group over an algebraically closed field. The largest cardinality of an algebraically independent set is called the transcendence degree of "L"/"K".