LAUSR

Let n ∈ ℤ with n ≥ 3. Let Sn and An denote, respectively, the symmetric group and alternating group on n letters. Let m be an indeterminate, and define fm(x) := xn + a(m,n)x + b(m,n), where a(m,n),b(m,n) are certain prescribed forms in m. For a certain set of these forms, we show unconditionally that there exist infinitely many primes p such that fp(x) is irreducible over ℚ, Galℚ(fp) = Sn, and the fields K = ℚ(????) are distinct and monogenic, where fp(????) = 0. Using a different set of forms, we establish a similar result for all square-free values of n ≡ 1(mod 4), with 5 ≤ n ≤ 401, and any positive integer value of m for which a(m,n) is square-free. Additionally, in this case, we prove that Galℚ(fp) = An. Finally, we show that these results can be extended under the assumption of the abc-conjecture. Our methods make use of recent results of Helfgott and Pasten.