Let Kn be a tamely ramified cyclic quintic field generated by a root of Emma Lehmer's parametric polynomial. We give all normal integral bases for Kn only by the roots of the polynomial, which is a generalization of the work of Lehmer in the case that n4+5n3+15n2+25n+25 is prime number, and Spearman-Willliams in the case that n4+5n3+15n2+25n+25 is square-free. Mathematics Subject Classification (2010) Primary 11R04 · 11R20, Secondary 11C08 · 11L05 · 11R80

Quandle homology theory has been developed and cocycles have been used to define invariants of oriented classical or surface links. We introduce a shifting chain map σ \sigma on each quandle chain complex that lowers the dimensions by one. By using its pull-back σ ♯ \sigma ^\sharp , each 2 2 -cocycle ϕ \phi gives us the 3 3 -cocycle σ ♯ ϕ \sigma ^\sharp \phi . For oriented classical links in the 3 3 -space, we explore relation between their quandle 2 2 -cocycle invariants associated with ϕ \phi and their shadow 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi . For oriented surface links in the 4 4 -space, we explore how powerful their quandle 3 3 -cocycle invariants associated with σ ♯ ϕ \sigma ^\sharp \phi are. Algebraic behavior of the shifting maps for low-dimensional (co)homology groups is also discussed.

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