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A family of real \(p^ n\)-tic fields. (English) Zbl 0834.11041

The authors construct a special family of polynomials \(P_n (X; a)\) of degree \(p^n\), where \(p\) is prime. The polynomials have coefficients in the ring of integers, say \({\mathcal O}_p\), of the \(p\)-th real cyclotomic field \(\mathbb{Q} (\zeta_p+ \zeta_p^{-1})\), and for each \(n\) they are irreducible except for finitely many values of \(a\). The zeros are all real and are permuted cyclically by a linear fractional transformation defined over \(\mathbb{Q} (\zeta_{p^n}+ \zeta^{-1}_{p^n})\). In the case \(p=3\) one has irreducibility without exceptions and \(P_n (X; a)\) can be regarded as a generalization of the “simplest” cubic polynomials of D. Shanks [Math. Comput. 28, 1137-1152 (1974; Zbl 0307.12005)]; note that \({\mathcal O}_3= \mathbb{Z}\).
The defining formula reads \(P_n (X; a)= R_n (X)- {a\over {p^n}} S_n (X)\), where \(a\in {\mathcal O}_p\) and \(R_n (X)\) and \(S_n (X)\) are polynomials in the expansion \((X- \zeta_p )^{p^n}= R_n (X)- \zeta_p S_n (X)\) (for \(p=2\), replace \(\zeta_p\) by \(\zeta_4\)). From the zeros of \(P_n (X; a)\) the authors obtain large (but, in general, non-maximal) sets of independent units in the splitting fields of these polynomials. – This generalizes the authors’ previous study of the case \(p=2\) (to appear).
The last section shows how the method can be modified to provide families of fields of degree \(p\) over \(\mathbb{Q} (\zeta_p+ \zeta_p^{-1})\), when \(p\) is composite.

MSC:

11R21 Other number fields
11R09 Polynomials (irreducibility, etc.)
11R18 Cyclotomic extensions
11R16 Cubic and quartic extensions
11R27 Units and factorization

Citations:

Zbl 0307.12005
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