A Family of Real 2 n -Tic Fields
Yuanyuan Shen, Lawrence C. Washington · Transactions of the American Mathematical Society · 1994
We study the family of polynomials P„(X; a) = n(( * + if)- ~nX + i)2") and determine when P„(X; a), a € 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"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> 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 = ±2 " and a = ±24 • 239. 1.