We now consider the case where , i.e when F corresponds to a complex cubic field. In this case the Hessian is an indefinite quadratic form, and in general there will be many reduced quadratic forms equivalent to it. Instead of using the Hessian, we will use a clever idea due to Matthews and Berwick (see [9]). If , then F has a unique real root , and if F is irreducible, so if we factor F in as

the quadratic form will be definite ( i.e ), but with real nonrational coefficients. We are going to show that the form has many of the properties of the Hessian.
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