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Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of computing a kind of particular basis of an ideal which enjoys some important properties. In this paper, we introduce the concept of Hadamard ideals in order to establish a new approach to the construction of Hadamard matrices with circulant core. Hadamard ideals reveal the rich interplay between Hadamard matrices with circulant core and ideals in multivariate polynomial rings. Hadamard ideals yield an exhaustive search for Hadamard matrices with circulant core for any specific dimension. In particular, we furnish all solutions for Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core. We establish the dihedral structure of the varieties associated with Hadamard ideals. Finally, we furnish the complete lists (exhaustive search) of inequivalent Hadamard matrices of the 12 orders 4, 8, \ldots, 44, 48 with circulant core.