Monogenic even cyclic sextic polynomials
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Lenny Jones
Abstract
Suppose that f(x) ∈ ℤ[x] is monic and irreducible over ℚ of degree N. We say that f(x) is monogenic if {1, θ, θ2, . . . , θN−1} is a basis for the ring of integers of ℚ(θ), where f(θ) = 0, and we say f(x) is cyclic if the Galois group of f(x) over ℚ is isomorphic to the cyclic group of order N. In this note, we prove that there do not exist any monogenic even cyclic sextic binomials or trinomials. Although the complete story on monogenic even cyclic sextic quadrinomials remains somewhat of a mystery, we nevertheless determine that the union of three particular infinite sets of cyclic sextic quadrinomials contains exactly four quadrinomials that are monogenic with distinct splitting fields. We also show that the situation can be quite different for quadrinomials whose Galois group is not cyclic.
Acknowledgement
The author thanks the anonymous referee for the helpful suggestions.
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(Communicated by István Gaál)
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Articles in the same Issue
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- Endomorphism kernel property for extraspecial and special groups
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