The solution demonstrates the existence of fields with 8 , 16 , and 25 elements by explicitly finding specific polynomials over their respective base fields (`ZZ_2` for 8 and 16 elements , `ZZ_5` for 25 elements).
What general mathematical principle ensures that we are always able to find such an irreducible polynomial of the required degree over any field `ZZ_p` , thus making it possible to construct any finite field of order `p^n`:
A) For every prime p and every positive integer n , there exists at least one irreducible polynomial of degree n over the field `Z_p`.
B) Only polynomials whose degrees are prime numbers (like 2 , 3 , or 5) can be irreducible over finite fields , simplifying the search for them ?