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Abstract A monic polynomial f (x) Zx of degree N is called monogenic if f (x) is irreducible over Q and \1, , ², , ^{N-1\} is a basis for the ring of integers of Q (), where f () =0. We prove that there exist exactly three distinct monogenic trinomials of the form x⁴+bx²+d whose Galois group is the cyclic group of order 4. We also show that the situation is quite different when the Galois group is not cyclic.
Lenny Jones (Fri,) studied this question.