Grobner bases are the primary tool for doing explicit computations in polynomial rings in many variables. In this book we give a leisurely introduction to the subject and its applications suitable for students with a little knowledge of abstract and linear algebra. The book contains not only the theory over fields, but also, the theory in modules and over rings.

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For a polynomial ring, R, in 4n variables over a field, we consider the submodule of R 4 corresponding to the 4 × 4n matrix made up of n groupings of the linear representation of quarternions with variable entries (which corresponds to the Cauchy-Fueter operator in partial differential equations) and let M n be the corresponding quotient module. We compute many homological properties of M n including the degrees of all of its syzygies, as well as its Betti numbers, Hilbert function, and dimension. We give similar results for its leading term module with respect to the degree reverse lexicographical ordering. The basic tool in the paper is the theory of Gröbner bases.