For an algebraic variety V over a field K, the dimension of V is the transcendence degree[?] over K of the function field[?] K(V) of all rational functions[?] on V, with values in K.
For the function field even to be defined, V here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the coordinate ring of V. It is easy to define by polynomials sets that have 'mixed dimension': a union of a curve and a plane in space, for example. These fail to be irreducible.
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