Abstract
In this chapter we focus on the ideals of the maximal order. We also discuss the ideals of any order of \(\mathbb{K}\), the lattices over \(\mathbb{K}\), and the properties of 1-lattices over \(\mathbb{K}\). We define the ideal class group of \(\mathbb{K}\) and the class number of \(\mathbb{K}\). We next examine the prime ideals in the maximal order and show that any non-zero ideal of \(\mathbb{K}\) can be represented uniquely as the product of prime ideals. We conclude with a review of the analytic class number formula and exhibit several results relating the class number of the cubic field to its regulator.
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