A triquadratic number field is a number field of degree 8 that is created by adjoining the square roots of three rational, squarefree integers to Q. We often denote a number field by K and its ring of integers by O_K. The ring of integers is defined to be the set of all elements of K that are zeros of a monic polynomial with coefficients in Z. It is already well-known that the ring of integers for a quadratic field Q(a) is given by all integer linear combinations of either {1, √a} or {1, (1+√a)/2 } depending on the value of a. These sets are called an integral basis for the ring of integers of the number field Q(a). The integral bases for the rings of integers for biquadratic fields are also already known.

In Chatelain’s paper he provides all of the information necessary to deter- mine the bases for all n-quadratic fields, but they are not presented in an explicit format. In this paper, we look at results of Chatelain and use these to determine more concise integral bases for triquadratic fields.