Editing Möbius function (section)

=== Algebraic number theory ===
Gauss{{sfn|Gauss|1986|loc=Art. 81}} proved that for a prime number <math>p</math> the sum of its [[Primitive root modulo n#Arithmetic facts|primitive roots]] is congruent to <math>\mu(p-1) \mod p</math>.

If <math>\mathbb{F}_q</math> denotes the [[finite field]] of order <math>q</math> (where <math>q</math> is necessarily a prime power), then the number <math>N</math> of monic irreducible polynomials of degree <math>n</math> over <math>\mathbb{F}_q</math> is given by{{sfn|Jacobson|2009|loc=§4.13}}

:<math>N(q,n)=\frac{1}{n} \sum_{d\mid n} \mu(d)q^\frac{n}{d}.</math>

The Möbius function is used in the [[Möbius inversion formula]].