  This paper explores the Tambara functor structure of the trace ideal of a
Galois extension. In the case of a (pro-)cyclic extension, we are able to
explicitly determine the generators of the ideal. Furthermore, we show that the
absolute trace ideal of a cyclic group is strongly principal when viewed as an
ideal of the Burnside Tambara Functor. Applying our results, we calculate the
trace ideal for extensions of finite fields. The appendix determines a formula
for the norm of a quadratic form over an arbitrary finite extension of a finite
field.
