ガロア第一論文と乗数イデアル他関連資料スレ17

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972: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2025/05/28(水) 08:17:23.55 ID:bsICkNCM

>>939
ID:CTASdXCp は、御大か
巡回ありがとうございます

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
果たして、合っているかどうか？　（＾＾

http://rio2016.5ch.net/test/read.cgi/math/1746597368/972

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