We study the irreducibility and the Galois group of the polynomial $f (a,x) = x^{8} +3ax^{6}+3a^{2}x^{4}+(a^{2}+1)ax^{2}+a^{2}+1$ over $\mathbf{Q}(a)$ and $\mathbf{Q}$. This polynomial is a factor of the 4-th dynatomic polynomial for the map $\sigma(x) = x^{3} + ax$.

We show that certain negatively twisted torus knots admit Dehn surgeries yielding 3-manifolds with non-left-orderable fundamental groups.

Let $(X,L)$ be a polarized manifold defined over the field of complex numbers. In this paper, we consider the case where $\dim X=4$ and we prove that the second Hilbert coefficient $A_{2}(X,L)$ of $(X,L)$, which was defined in our previous paper, is non-negative. Furthermore we consider a question proposed by H. Tsuji for $\dim X=4$.