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This paper deals with some special integral transforms in the setting of quaternionic valued slice polyanalytic functions. In particular, using the polyanalytic Fueter mappings, it is possible to construct a new family of polynomials which are called the generalized Appell polynomials. Furthermore, the range of the polyanalytic Fueter mappings on two different polyanalytic Fock spaces is characterized. Finally, we study the polyanalytic Fueter–Bargmann transforms. [ABSTRACT FROM AUTHOR]

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(𝜃) are distinct and monogenic, where fp(𝜃)=0. Using a different set of forms, we establish a similar result for all square-free values of n≡1(mod4), 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. [ABSTRACT FROM AUTHOR]

An algebraic number ring is monogenic, or one-generated, if it has the form for a single algebraic integer . It is a problem of Hasse to characterize, whether an algebraic number ring is monogenic or not. In this note, we prove that if is a square-free rational integer, and , then the pure sextic field is not monogenic. Our results are illustrated by examples. [ABSTRACT FROM AUTHOR]