A family of real $2^ n$-tic fields

Abstract:

We study the family of polynomials \[ {P_n}(X;a) = \Re ({(X + i)^{{2^n}}}) - \frac {a}{{{2^n}}}\Im ({(X + i)^{{2^n}}})\] and determine when ${P_n}(X;a)$, $a \in \mathbb {Z}$, is irreducible. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the ${2^n}$th cyclotomic field. The families of fields we obtain are natural extensions of those studied by M.-N. Gras and Y.-Y. Shen, but in general the present fields are non-Galois for $n \geq 4$. From the roots we obtain a set of independent units for the Galois closure that generate an "almost fundamental piece" of the full group of units. Finally, we discuss the two examples where our fields are Galois, namely $a = \pm {2^n}$ and $a = \pm {2^4} \bullet 239$.