ABSTRACT
We are concerned with the interrelationship between two basic objects in algebraic number theory: the ring of integers and the Galois group. In particular, we seek to understand the effect of the Galois group upon the ring of integers. At the same time, we are also interested in the Galois action upon other fractional ideals. So that the action may be similar, we restrict ourselves to ambiguous ideals – those that are mapped to themselves by the Galois group. The setting for our investigation is the family of C8-extensions. This choice is guided by by a result of E. Noether as well as results in Integral Representation Theory. Noether’s Normal Integral Basis Theorem. A finite Galois extension of number fields N/K is said to be at most tamely ramified (TAME) if the factorization of each prime ideal PK (of OK) in ON results in exponents (degrees of ramification) that are relatively prime to the ideal PK . A normal integral basis (NIB) is said to exist if there is an element α ∈ ON (in the ring of integers of N) whose conjugates, {σα : σ ∈ Gal(N/K)}, provide a basis for ON over OK (the integers in K).
