Minkowski proved that the ideal class group is finite using the Minkowski map to turn the problem into a geometric one. He showed that under this map, the image of any integral basis for the ring of integers is contained in the fundamental parallelotope, a set with volume a constant multiple of the square root of the discriminant.

This gives rise to the following question: How large does the volume of a set need to be so that the set contains the image of an integral basis? This paper will partially answer this question by showing in a poly-quadratic field, there is a set with volume a constant multiple of the discriminant that does not contain the image of any integral basis. Most of this paper comes from the research project that I participated in the Budapest Semester in Mathematics last summer.