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Based on the isomorphic algebraic structures of the 2\( \mathbb{D} \) Euclidean field of complex vectors \( \mathbf{V}_{\mathbb{C}} \) and the field of complex numbers \( \mathbb{C} \), in terms of identical geometric products of the elements of both fields, this paper brings the algebraic structure of a 3\( \mathbb{D} \) field of complex vectors, as well as the corresponding fundamental integral identities in those vector fields.