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

In this paper, we characterize whether the pure sextic fields Q( 6 √ m) with square-free integers m ≡ ±1 (mod 9) have power integral bases or do not; if m ≡ 2, 3 (mod 4), then Q( 6 √ m) have power integral bases. We prove this by determining relative integral bases of such fields with respect to their cubic and quadratic subfields. Based on the works of Kovács and Pethő, several examples on application of monogenic fields to CNS (Canonical Number System) are shown.

Pythagorean fuzzy soft set (PFSS) is the most influential and operative extension of the Pythagorean fuzzy set (PFS), which contracts with the parametrized standards of the substitutes. It is also a generalized form of the intuitionistic fuzzy soft set (IFSS) and delivers a well and accurate estimation in the decision-making (DM) procedure. The primary purpose is to prolong and propose ideas related to Einstein’s ordered weighted aggregation operator from fuzzy to PFSS, comforting the condition that the sum of the degrees of membership function and nonmembership function is less than one and the sum of the squares of the degree of membership function and nonmembership function is less than one. We present a novel Pythagorean fuzzy soft Einstein ordered weighted averaging (PFSEOWA) operator based on operational laws for Pythagorean fuzzy soft numbers. Furthermore, some essential properties such as idempotency, boundedness, and homogeneity for the proposed operator have been presented in detail. The choice of a sustainable supplier is also examined as an essential part of sustainable supply chain management (SSCM) and is considered a crucial multiattribute group decision-making (MAGDM) issue. In some MAGDM problems, the relationship between alternatives and uncertain environments will be the main reason for deficient consequences. We have presented a novel aggregation operator for PFSS information to choose sustainable suppliers to cope with those complex issues. The Pythagorean fuzzy soft number (PFSN) helps to represent the obscure information in such real-world perspectives. The priority relationship of PFSS details is beneficial in coping with SSCM. The proposed method’s effectiveness is proved by comparing advantages, effectiveness, and flexibility among the existing studies.

We have developed a rigorous computational technique to compute exact analytic expressions for a number of distance-based topological indices of chemical graphs. There are two main advantages of our technique over existing techniques of similar nature: rst, our technique is signicantly diverse as it also covers the Wiener index and eccentricitybased topological indices besides Szeged-like indices, and secondly we have considerably reduced the algorithmic and computational complexity in comparison to previous techniques. Our proposed technique generates certain vertex and edge partitions of a graph which are essential in computing the exact analytical formulas of distance-based and eccentricity-based indices. To ensure the applicability of our technique, we have computed various distance-based and eccentricity-based topological indices for certain innite families of polyomino chain system. Moreover, we nd analytical exact expressions of certain degree-based topological indices for these polyomino chains. These topological indices can be obtained as a by-product of our technique.

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