We begin this chapter by considering another application of Mobius inversion, this time to the result of Theorem 6.1, viz. $$ {x^{{{q^n}}}} - x = \prod\limits_{{d|n}} {{V_d}} (x) $$ , where V d (x) denotes the product of all monic irreducible polynomials of degree d over GF(q). We recall (see Example 6.3) that here the underlying group is the set of rational functions \( \{ {\text{p}}\left( {\text{x}} \right)/{\text{q}}\left( {\text{x}} \right):{\text{p}}\left( {\text{x}} \right),{\text{q}}\left( {\text{x}} \right) \in {\text{k}}\left[ {\text{x}} \right],{\text{p}}\left( {\text{x}} \right) \ne 0,{\text{g}}\left( {\text{x}} \right) \ne 0\} \), and the group operation is multiplication. When the group is written multiplicatively, Theorem 6.3 becomes $$ b(n) = \prod\limits_{{d|n}} {a{{(d)}^{{\mu (n/d)}}}} $$ (7.1) , where a0 = 1 (the group’s identity), a+1 = a, a-1 = the inverse of a. Thus combining (7.1) with Theorem 6.1, we obtain $$ {V_n}(x) = \prod\limits_{{d|n}} {({x^{{{q^d}}}}} - x{)^{{\mu (n/d)}}} $$ (7.2) .