Carmichael numbers in number rings Original Research Article

We generalize Carmichael numbers to ideals in number rings and prove a generalization of Korseltʹs Criterion for these Carmichael ideals. We investigate when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of abelian number fields. In particular, we show that given any composite integer n, there exist infinitely many quadratic number fields in which n is not Carmichael. Finally, we show that there are infinitely many abelian number fields K with discriminant relatively prime to n such that n is not Carmichael in K.