Quinto esercizio Esercizio 7.5 Siano p , q {\displaystyle p,q} primi distinti. Esprimere ϕ p q ( x ) {\displaystyle \phi _{pq}(x)} in termini di ϕ p ( x ) {\displaystyle \phi _{p}(x)} . Per il primo lemma, siccome gli unici divisori del prodotto p q {\displaystyle pq} sono 1 , p , q , p q {\displaystyle 1,p,q,pq} , si ha ϕ p q ( x ) = x p q − 1 ϕ 1 ( x ) ∗ ϕ p ( x ) ∗ ϕ q ( x ) {\displaystyle \phi _{pq}(x)={\frac {x^{pq}-1}{\phi _{1}(x)*\phi _{p}(x)*\phi _{q}(x)}}} ϕ p q ( x ) = x p q − 1 ϕ p ( x ) ∗ ( x q − 1 ) {\displaystyle \phi _{pq}(x)={\frac {x^{pq}-1}{\phi _{p}(x)*(x^{q}-1)}}} Moltiplico e divido per ϕ 1 ( x ) = x − 1 {\displaystyle \phi _{1}(x)=x-1} : ϕ p q ( x ) = ( x p q − 1 ) ( x − 1 ) ( x − 1 ) ∗ ϕ p ( x ) ∗ ( x q − 1 ) {\displaystyle \phi _{pq}(x)={\frac {(x^{pq}-1)(x-1)}{(x-1)*\phi _{p}(x)*(x^{q}-1)}}} ϕ p q ( x ) = ( x p q − 1 ) ( x − 1 ) ( x p − 1 ) ∗ ( x q − 1 ) {\displaystyle \phi _{pq}(x)={\frac {(x^{pq}-1)(x-1)}{(x^{p}-1)*(x^{q}-1)}}} ϕ p q ( x ) = ( ( x q ) p − 1 ) ( x − 1 ) ( x p − 1 ) ∗ ( x q − 1 ) {\displaystyle \phi _{pq}(x)={\frac {((x^{q})^{p}-1)(x-1)}{(x^{p}-1)*(x^{q}-1)}}} ϕ p q ( x ) = ( x q ) p − 1 x q − 1 ∗ x − 1 x p − 1 {\displaystyle \phi _{pq}(x)={\frac {(x^{q})^{p}-1}{x^{q}-1}}*{\frac {x-1}{x^{p}-1}}} = ϕ p ( x q ) ϕ p ( x ) {\displaystyle ={\frac {\phi _{p}(x^{q})}{\phi _{p}(x)}}}