Noether's problem for some subgroups of S14: The modular case

Hang Fu; Ming chang Kang; Baoshan Wang; Jian Zhou

doi:10.1016/j.jalgebra.2020.10.019

Noether's problem for some subgroups of S₁₄: The modular case

Hang Fu^*
, Ming chang Kang
, Baoshan Wang
, Jian Zhou

^*此作品的通讯作者

数学科学学院

National Taiwan University
Peking University

科研成果: 期刊稿件 › 文章 › 同行评审

摘要

Let G be a subgroup of S_n, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x₁,⋯,x_n) via k-automorphisms defined by σ⋅x_i:=x_σ⋅i for any σ∈G and 1≤i≤n. In this article, we will show that if G is a solvable transitive subgroup of S₁₄ and char(k)=7, then the fixed subfield k(x₁,⋯,x₁₄)^G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gaschütz Theorem or some ideas in its proof.

源语言	英语
页（从-至）	529-546
页数	18
期刊	Journal of Algebra
卷	568
DOI	https://doi.org/10.1016/j.jalgebra.2020.10.019
出版状态	已出版 - 15 2月 2021

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10.1016/j.jalgebra.2020.10.019

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引用此

@article{50dad88999004c698c27019e41761bad,

title = "Noether's problem for some subgroups of S14: The modular case",

abstract = "Let G be a subgroup of Sn, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x1,⋯,xn) via k-automorphisms defined by σ⋅xi:=xσ⋅i for any σ∈G and 1≤i≤n. In this article, we will show that if G is a solvable transitive subgroup of S14 and char(k)=7, then the fixed subfield k(x1,⋯,x14)G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gasch{\"u}tz Theorem or some ideas in its proof.",

keywords = "Gasch{\"u}tz's Theorem, Kuniyoshi's Theorem, L{\"u}roth's problem, Noether's problem, Rationality problem",

author = "Hang Fu and Kang, \{Ming chang\} and Baoshan Wang and Jian Zhou",

note = "Publisher Copyright: {\textcopyright} 2020 Elsevier Inc.",

year = "2021",

month = feb,

day = "15",

doi = "10.1016/j.jalgebra.2020.10.019",

language = "英语",

volume = "568",

pages = "529--546",

journal = "Journal of Algebra",

issn = "0021-8693",

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TY - JOUR

T1 - Noether's problem for some subgroups of S14

T2 - The modular case

AU - Fu, Hang

AU - Kang, Ming chang

AU - Wang, Baoshan

AU - Zhou, Jian

PY - 2021/2/15

Y1 - 2021/2/15

N2 - Let G be a subgroup of Sn, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x1,⋯,xn) via k-automorphisms defined by σ⋅xi:=xσ⋅i for any σ∈G and 1≤i≤n. In this article, we will show that if G is a solvable transitive subgroup of S14 and char(k)=7, then the fixed subfield k(x1,⋯,x14)G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gaschütz Theorem or some ideas in its proof.

AB - Let G be a subgroup of Sn, the symmetric group of degree n. For any field k, G acts naturally on the rational function field k(x1,⋯,xn) via k-automorphisms defined by σ⋅xi:=xσ⋅i for any σ∈G and 1≤i≤n. In this article, we will show that if G is a solvable transitive subgroup of S14 and char(k)=7, then the fixed subfield k(x1,⋯,x14)G is rational (i.e., purely transcendental) over k. In proving the above theorem, we rely on the Kuniyoshi–Gaschütz Theorem or some ideas in its proof.

KW - Gaschütz's Theorem

KW - Kuniyoshi's Theorem

KW - Lüroth's problem

KW - Noether's problem

KW - Rationality problem

UR - https://www.scopus.com/pages/publications/85095775112

U2 - 10.1016/j.jalgebra.2020.10.019

DO - 10.1016/j.jalgebra.2020.10.019

M3 - 文章

AN - SCOPUS:85095775112

SN - 0021-8693

VL - 568

SP - 529

EP - 546

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Noether's problem for some subgroups of S14: The modular case

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Noether's problem for some subgroups of S₁₄: The modular case