It is known that non-isomorphic number fields can share the same Dedekind zeta function. However, there don't appear to be any examples of very low degree so in these cases the zeta function must determine the field. How hard is it to show this?
Questions: (1) If K is a cubic extension of the rationals, how can one recover$\,$ K$\,$ from $\,$$\zeta_K$(s) ?
$\quad$$\quad$$\quad$$\quad$$\,$(2) Are there any near misses where one zeta function is almost but not quite equal to another? That is, if two Dedekind zeta functions (expanded as Dirichlet series) agree for all but finitely many terms, must they be identical?
Thanks.