ガロア第一論文と乗数イデアル他関連資料スレ18

ガロア第一論文と乗数イデアル他関連資料スレ18 (541ﾚｽ)
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26: 現代数学の系譜雑談 ◆yH25M02vWFhP 05/28(水)14:52 ID:vzADU7Bh(4/6) AA×

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26: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/05/28(水) 14:52:45.07 ID:vzADU7Bh

前スレ フォロー
https://rio2016.5ch.net/test/read.cgi/math/1746597368/972
(引用開始)
２と３と５と７の平方根が 有理数体上線形独立であることを 初めて示したのは誰？
　↓英訳
Who first showed that the square roots of 2, 3, 5, and 7 are linearly independent over the field of rational numbers?
Copilotさんに喰わせると
その答え：
The linear independence of square roots of distinct square-free integers over ℚ has been studied extensively in number theory and algebra. One of the earliest rigorous treatments of this topic can be traced back to A.S. Besicovitch (1940), who explored the linear independence of fractional powers of integers. Later, L.J. Mordell (1953) also contributed to the study of the linear independence of algebraic numbers.
The general result states that if n₁, n₂, ..., nₖ are distinct square-free integers, then {√n₁, √n₂, ..., √nₖ} is linearly independent over ℚ. This follows from deeper results in Kummer theory and the Galois theory of radical extensions.
Would you like to explore the proof in more detail? It involves some fascinating applications of field extensions and trace arguments.
でした
(引用終り)

さて、フォローです
google検索：L.J. Mordell (1953) Kummer theory the square roots of 2, 3, 5, and 7 are linearly independent over the field of rational numbers
で、下記ヒット
Besicovitchと Mordellと 両方 stackexchangeの回答にあります（これがネタ？）

なお、arxiv投稿 ”ON THE DEGREE OF REPEATED RADICAL EXTENSIONS FERNANDO SZECHTMAN”2021
が、本格的です（Can. Math. Bull. 64 (2021) 877-885 に掲載か）

(参考)
https://math.stackexchange.com/questions/30687/the-square-roots-of-different-primes-are-linearly-independent-over-the-field-of
The square roots of different primes are linearly independent over the field of rationals
asked Apr 3, 2011 at user8465
(抜粋)
answered Apr 3, 2011 at Bill Dubuque

Elementary proofs like that above are often credited to Besicovitch (see below). But I have not seen his paper so I cannot say for sure whether or not Besicovic's proof is essentially the same as above. Finally, see the papers reviewed below for some stronger results.

2,33f 10.0X
Besicovitch, A. S.
On the linear independence of fractional powers of integers.
J. London Math. Soc. 15 (1940). 3-6.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1748354585/26

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