Editing Carmichael number (section)

== Generalizations ==
The notion of Carmichael number generalizes to a Carmichael ideal in any [[number field]] {{tmath|1= K }}. For any nonzero [[prime ideal]] <math>\mathfrak p</math> in {{tmath|1= {\mathcal O}_K }}, we have <math>\alpha^{{\rm N}(\mathfrak p)} \equiv \alpha \bmod {\mathfrak p}</math> for all <math>\alpha</math> in {{tmath|1= {\mathcal O}_K }}, where <math>{\rm N}(\mathfrak p)</math> is the norm of the [[Ideal (ring theory)|ideal]] {{tmath|1= \mathfrak p }}. (This generalizes Fermat's little theorem, that <math>m^p \equiv m \bmod p</math> for all integers {{tmath|1= m }} when {{tmath|1= p }} is prime.) Call a nonzero ideal <math>\mathfrak a</math> in <math>{\mathcal O}_K</math> Carmichael if it is not a prime ideal and <math>\alpha^{{\rm N}(\mathfrak a)} \equiv \alpha \bmod {\mathfrak a}</math> for all {{tmath|1= \alpha \in {\mathcal O}_K }}, where <math>{\rm N}(\mathfrak a)</math> is the norm of the ideal {{tmath|1= \mathfrak a }}.  When {{tmath|1= K }} is {{tmath|1= \mathbf Q }}, the ideal <math>\mathfrak a</math> is [[Principal ideal|principal]], and if we let {{tmath|1= a }} be its positive generator then the ideal <math>\mathfrak a = (a)</math> is Carmichael exactly when {{tmath|1= a }} is a Carmichael number in the usual sense.

When {{tmath|1= K }} is larger than the [[Rational number|rational]]s it is easy to write down Carmichael ideals in {{tmath|1= {\mathcal O}_K }}: for any prime number {{tmath|1= p }} that splits completely in {{tmath|1= K }}, the principal ideal <math>p{\mathcal O}_K</math> is a Carmichael ideal. Since infinitely many prime numbers split completely in any number field, there are infinitely many Carmichael ideals in {{tmath|1= {\mathcal O}_K }}. For example, if {{tmath|1= p }} is any prime number that is 1 mod 4, the ideal {{tmath|1= (p) }} in the [[Gaussian integer]]s <math>\mathbb Z[i]</math> is a Carmichael ideal.

Both prime and Carmichael numbers satisfy the following equality:
: <math>\gcd \left(\sum_{x=1}^{n-1} x^{n-1}, n\right) = 1.</math>