## LSU Historical Dissertations and Theses

#### Title

A New Condition for Arithmetic Equivalence.

1991

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Robert V. Perlis

#### Abstract

Two algebraic number fields K, K$\sp\prime$ are said to be arithmetically equivalent if every prime number p has the same splitting type in K as in K$\sp\prime$. Many equivalent formulations of arithmetic equivalence are known. The best-known of these is: K and K$\sp\prime$ are arithmetically equivalent if and only if their Dedekind zeta functions coincide. This dissertation provides a surprising new formulation: K and K$\sp\prime$ are arithmetically equivalent if and only if for all prime numbers p, outside a given set of density zero, the number $g\sb K (p)$ of prime factors of p in K is equal to the number $g\sb {K\sp\prime} (p)$ of prime factors of p in K$\sp\prime$. This is of interest because it makes reference only to the splitting numbers $g\sb {K} (p)$, $g\sb {K\sp\prime} (p)$ and not to the inertia degrees of the prime factors of p involved. This theorem has many non-trivial consequences.

31

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