Irreducible polynomials over GF(5)
Jelly, 30 23 22 20 bytes
ÆF&gt;1’PḄ
ÆDµU5*×Ç€S:Ṫ

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Algorithm
This uses the formula
$$\text{A001692}(n) = \frac 1 n \sum_{d|n} \mu(d)5^\frac n d$$
from the OEIS page, where \$d | n\$ indicates that we sum over all divisors \$d\$ of \$n\$, and \$\mu\$ represents the Möbius function.
Code
ÆF&gt;1’PḄ       Monadic helper link. Argument: d
              This link computes the Möbius function of d.

ÆF            Factor d into prime-exponent pairs.
  &gt;1          Compare each prime and exponent with 1. Returns 1 or 0.
    ’         Decrement each Boolean, resulting in 0 or -1.
     P        Take the product of all Booleans, for both primes and exponents.
      Ḅ       Convert from base 2 to integer. This is a sneaky way to map [0, b] to
              b and [] to 0.

ÆDµU5*×Ç€S:Ṫ  Main link. Input: n

ÆD            Compute all divisors of n.
  µ           Begin a new, monadic chain. Argument: divisors of n
   U          Reverse the divisors, effectively computing n/d for each divisor d.
    5*        Compute 5 ** (n/d) for each n/d.

       Ç€     Map the helper link over the (ascending) divisors.
      ×       Multiply the powers by the results from Ç.
         S    Add the resulting products.
          :Ṫ  Divide the sum by the last divisor (n).