The polynomial F_p(x) = x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.
Jones, Monogenic even octic polynomials and their Galois groups, New York J
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Wieferich Primes and Monogenic Trinomials
The polynomial F_p(x) = x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.