Is there a simple way to find an n-1 degree polynomial with discriminant D = n^k, (defined with one variable x) and n is prime?

I am not talking about the polynomials of the (simple) form(s) (a^n-b^n)/(a-b) or (a^n+b^n)/(a+b) as those polynomial forms are excluded from this thread, they can be expanded into n-1 degree polynomials already.

The property I am getting at is the n-1 degree polynomial has factors either 0 or 1 (mod n) when the polynomial is evaluated for any "x" value.

Some examples for n = 3 (which don't have the simple form(s) (a^3-b^3)/(a-b) or (a^3+b^3)/(a+b))

19x^2+7x+1

37x^2+11x+1

.....

Which come from the discriminant D = b^2-4ac for ax^2+bx+c.

Similarly, WITHOUT searching for coefficients, are there good examples of degree 4, discriminant a power of 5, and of degree 6, discriminant a power of 7. Again, these polynomial must not have any of the Generalized Mersenne form(s) (a^n-b^n)/(a-b) and (a^n+b^n)/(a+b)

The discriminant D is at:

For degree 4, integer solutions (a, b, c, d, e) such that:

256 a^3 e^3 - 192 a^2 b d e^2 - 128 a^2 c^2 e^2 + 144 a^2 c d^2 e - 27 a^2 d^4 + 144 a b^2 c e^2 - 6 a b^2 d^2 e - 80 a b c^2 d e + 18 a b c d^3 + 16 a c^4 e - 4 a c^3 d^2 - 27 b^4 e^2 + 18 b^3 c d e - 4 b^3 d^3 - 4 b^2 c^3 e + b^2 c^2 d^2 = 5^n

Computed

here.

For degree 6, integer solutions (a, b, c, d, e, f, g) such that:

Too long for thread, is

here.

For higher goals, for all primes n < 1000, (which is extremely complicated to do). Like always, hints, suggestions, and answers are appreciated.