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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

A family of real $ 2\sp n$-tic fields


Authors: Yuan Yuan Shen and Lawrence C. Washington
Journal: Trans. Amer. Math. Soc. 345 (1994), 413-434
MSC: Primary 11R21; Secondary 11R09, 11R27
MathSciNet review: 1264151
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Abstract: We study the family of polynomials

$\displaystyle {P_n}(X;a) = \Re ({(X + i)^{{2^n}}}) - \frac{a}{{{2^n}}}\Im ({(X + i)^{{2^n}}})$

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$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1264151-6
PII: S 0002-9947(1994)1264151-6
Keywords: $ {2^n}$-tic fields, units
Article copyright: © Copyright 1994 American Mathematical Society