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On a Theorem of Childs on Normal Bases of Rings of Integers
Author(s) -
Ichimura Humio
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070300440x
Subject(s) - mathematics , field (mathematics) , set (abstract data type) , algebraic number field , prime (order theory) , quadratic field , degree (music) , prime number , discrete mathematics , quadratic equation , combinatorics , pure mathematics , quadratic function , physics , computer science , geometry , acoustics , programming language
Let p be a prime number, F a number field, andH nthe set of all unramified cyclic extensions over F of degree p having a relative normal integral basis. When ζ p ∈ F × , Childs determined the setH nin terms of Kummer generators. When p =3 and F is an imaginary quadratic field, Brinkhuis determined this set in a form which is, in a sense, analogous to Childs's result. The paper determines this set for all p ⩾ 3 and F with ζ p ∉ F × (and satisfying an additional condition), using the result of Childs and a technique developed by Brinkhuis. Two applications are also given.