A Family of RealPn-Tic Fields
Yuan-Yuan Shen, Lawrence C. Washington · Canadian Journal of Mathematics · 1995
Abstract Letq=pifpis an odd prime,q= 4 ifp= 2. Let ζqbe any primitiveq-th root of unity, and let . We study the family of polynomials where Rn(X) and Sn(X) are the polynomials in the expansion We show that for fixedn,Pn(X;a) is irreducible for all but finitely many a ∈ O, and forp= 3, we show that it is irreducible for all a ∈ O. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real pn-th cyclotomic field. From the roots we obtain a non-maximal set of independent units for the splitting field. In the last section we briefly treat extensions of our methods to compositep.