On the reducibility of X q − 1 in the monoid ring F p [ X ; 〈 2 , 3 〉 ]

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Using techniques of algebraic and analytic number theory, we resolve a question on monoid rings posed by Kulosman et al. under the assumption of the Generalized Riemann Hypothesis (GRH). Specifically, we show that under an appropriate GRH, for any (rational) prime p the set E ( p ) = { q prime | X q − 1 factors in F p [ X ; M ] } , where M = 〈 2 , 3 〉 = N 0 ∖ { 1 } , contains a subset with positive natural density. In particular E ( p ) ≠ ∅ . This proves that M is not a so-called “Matsuda monoid” of any positive type. For p = 2 , 3 this was observed by Kulosman, who provided factorizations of X 7 − 1 and X 11 − 1 in F 2 [ X ; M ] and F 3 [ X ; M ] , respectively. Our results explain and reproduce both of these factorizations, as well.