论文标题
关于某些纯度元环节20度20的投降问题
On the capitulation problem of some pure metacyclic fields of degree 20
论文作者
论文摘要
令$γ\，= \，\ Mathbb {q}（\ sqrt [5] {n}）$是一个纯Quintic字段，其中$ n $是一个正整数$ 5^{th} $ power-power，$ k_0 \，$ k_0 \，= \，= \，\ mathbb {q} Q}（q} Q}（Q}（Q} ecclot cyclot cyclot cyclot cyclot， $ 5^{th} $ unity $ζ_5$的根和$ k \，= \，\ Mathbb {q}（\ sqrt [5] {n}，ζ_5）$ $γ$的正常闭合。令$ k_5^{（1）} $为希尔伯特$ 5 $ - $ k $，$ c_ {k，5} $ $ 5 $ - 理想的$ k $的$ 5 $ - ideal类组，以及$ c_ {k，5}^{（σ）} $在$ gal（k/k_0）$ gal（k__0）$ gal（k__0）的歧义类中。当$ c_ {k，5} $是类型$（5,5）$和等级$ c_ {k，5}^{（σ）} \，= \，1 $时，我们研究了$ 5 $ -IDEAL类的投票问题$ C_ {k，5} $的$ 5 $ C_ {K，5} $，在$ k_5^^^^$ K_5^^{1）中。
Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a positive integer $5^{th}$ power-free, $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $ζ_5$, and $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ the normal closure of $Γ$. Let $k_5^{(1)}$ be the Hilbert $5$-class field of $k$, $C_{k,5}$ the $5$-ideal classes group of $k$, and $C_{k,5}^{(σ)}$ the group of ambiguous classes under the action of $Gal(k/k_0)$ = $\langleσ\rangle$. When $C_{k,5}$ is of type $(5,5)$ and rank $C_{k,5}^{(σ)}\,=\,1$, we study the capitulation problem of the $5$-ideal classes of $C_{k,5}$ in the six intermediate extensions of $k_5^{(1)}/k$.
