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981: １３２人目の素数さん [sage] 2024/06/12(水) 06:33:21.36 ID:kQ+gVNo/(1/3) AAS
>>979
Let
𝐾
be a field and
𝑀
be an intermediate field between
𝐾
and
𝐾(𝑋)
for some indeterminate X. Then there exists a rational function
𝑓(𝑋)∈𝐾(𝑋)
such that
𝑀=𝐾(𝑓(𝑋)).
In other words, every intermediate extension between
𝐾 and 𝐾(𝑋)
is a simple extension.

Proofs
The proof of Lüroth's theorem can be derived easily from the theory of rational curves, using the geometric genus. This method is non-elementary, but several short proofs using only the basics of field theory have long been known, mainly using the concept of transcendence degree. Many of these simple proofs use Gauss's lemma on primitive polynomials as a main step.

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983(1): １３２人目の素数さん [sage] 2024/06/12(水) 08:27:55.68 ID:kQ+gVNo/(2/3) AAS
柳老面

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986: １３２人目の素数さん [sage] 2024/06/12(水) 18:26:38.75 ID:kQ+gVNo/(3/3) AAS
Rudinの本が有名なのは
最初に指数関数を用いたWeierstrassの
円周率の定義が書いてあるから

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