My initial question in the present post is pretty basic:
Is the (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ a square for integers $k \neq 1$ and $k \equiv 1 \pmod 4$, and primes $p \equiv 1 \pmod 4$?
When $k=1$, I know that this polynomial is equal to $$p^4 - 4p^3 + 6p^2 - 4p + 1 = (p - 1)^4$$ which is a square. I am interested in the cases when $k \neq 1$ and $k \equiv 1 \pmod 4$.
MY ATTEMPT
Already, when $k=5$, we have the rather curious-looking factorization $$p^{12} - 4p^7 + 6p^6 - 4p^5 + 1$$ $$= (p - 1)^2 \cdot \bigg(p^{10} + 2p^9 + 3p^8 + 4p^7 + 5p^6 + 2p^5 + 5p^4 + 4p^3 + 3p^2 + 2p + 1\bigg).$$
So I am led to conjecture that:
CONJECTURE A: The (integer) polynomial $$p^{2k+2} - 4p^{k+2} + 6p^{k+1} - 4p^k + 1$$ is not a square for $k \neq 1$ and $k \equiv 1 \pmod 4$.
Alas, I have no proof for this conjecture.
FINAL QUESTION Can you prove Conjecture A?