@inbook{a49eedd219594db0986cce13fd722646,

title = "Lower Bounds for Heights in Relative Galois Extensions",

abstract = "The goal of this paper is to obtain lower bounds on the height of an algebraic number in a relative setting, extending previous work of Amoroso and Masser. Specifically, in our first theorem, we obtain an effective bound for the height of an algebraic number {\^I}± when the base field {\dh}�•‚ is a number field and {\dh}�•‚({\^I}±) {\^a}ˆ• {\dh}�•‚ is Galois. Our second result establishes an explicit height bound for any nonzero element {\^I}± which is not a root of unity in a Galois extension {\dh}�”½{\^a}ˆ• {\dh}�•‚, depending on the degree of {\dh}�•‚{\^a}ˆ• {\^a}„{\v s} and the number of conjugates of {\^I}± which are multiplicatively independent over {\dh}�•‚. As a consequence, we obtain a height bound for such {\^I}± that is independent of the multiplicative independence condition.",

author = "Shabnam Akhtari and Kevser Akta{\AA}Ÿ and Biggs, {Kirsti D.} and Alia Hamieh and Kathleen Petersen and Lola Thompson",

note = "Publisher Copyright: {\textcopyright} 2018, The Author(s) and the Association for Women in Mathematics.",

year = "2018",

doi = "10.1007/978-3-319-74998-3_1",

language = "English (US)",

series = "Association for Women in Mathematics Series",

publisher = "Springer",

pages = "1--17",

booktitle = "Association for Women in Mathematics Series",

}