( x − α k ). (i) Show that, for every integer n ≥ 1, the product of all monic irreducible polynomials over F q whose degrees divide n is equal to x q n − x . (ii) Let I q ( d ) denote the number of...

(

x



−

α



_k






).

1. 
   

   
   (i) Show that, for every integer
   
   n
   
   ≥ 1, the product of all monic
   

   
   

irreducible polynomials over

F



_q


whose degrees divide

n

is equal

to

x



^q


n

−

x



.

(ii) Let

I



_q


(

d

) denote the number of monic irreducible polynomials of

degree

d

in

F



_q


[

x

]. Show that

q



ⁿ






=

⁾




d




I



_q






(

d

) for all

n



∈

N


,

d

|

n

where the sum is extended over all positive divisors

d

of

n

.