Tables of pure quintic fields

纯五次域的单生成性：筛法的力量

Monogenity of pure quintic fields: the power of sieving

本文提出一种简单算法，用于计算纯五次域中的幂积分基生成器，通过改进的筛法显著减少需考虑的小指数数量。

On the exact divisibility by $5$ of the class number of some pure metacyclic fields

Let $Γ\,=\, \mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a natural number $5^{th}$ power-free. Let $k = \mathbb{Q}(\sqrt[5]{n}, ζ_5)$, with $ζ_5$ is a primitive $5^{th}$ root of unit, be the normal closure of $Γ$, and a pure metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some particular forms, we show that $Γ$ admits a trivial $5$-class group and $5$ divides exactly the class number of $k$.

On the capitulation problem of some pure metacyclic fields of degree 20 II

Let $n$ be a $5^{th}$ power-free naturel number and $k_0\,=\,\mathbb{Q}(ζ_5)$ be the cyclotomic field generated by a primitive $5^{th}$ root of unity $ζ_5$. Then $k\,=\,\mathbb{Q}(\sqrt[5]{n},ζ_5)$ is a pure metacyclic field of absolute degree $20$. In the case that $k$ possesses a $5$-class group $C_{k,5}$ of type $(5,5)$ and all the classes are ambiguous under the action of $Gal(k/k_0)$, the capitulation of $5$-ideal classes of $k$ in its unramified cyclic quintic extensions is determined.

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

In this paper, we determine mod $2$ Galois representations $\barρ_{ψ,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5ψX_0X_1X_2X_3X_4=0,\ ψ\in K$$ defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_ψ$ below. Applying this result, when $K=F$ is a totally real field, for some at most qaudratic totally real extension $M/F$, we prove that $\barρ_{ψ,2}|_{G_M}$ is associated to a Hilbert-Siegel modular H…

On the capitulation problem of some pure metacyclic fields of degree 20

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…

Differential principal factors and Polya property of pure metacyclic fields

Barrucand and Cohn's theory of principal factorizations in pure cubic fields \(\mathbb{Q}(\sqrt[3]{D})\) and their Galois closures \(\mathbb{Q}(ζ_3,\sqrt[3]{D})\) with \(3\) types is generalized to pure quintic fields \(L=\mathbb{Q}(\sqrt[5]{D})\) and pure metacyclic fields \(N=\mathbb{Q}(ζ_5,\sqrt[5]{D})\) with \(13\) possible types. The classification is based on the Galois cohomology of the unit group \(U_N\), viewed as a module over the automorphism group \(\mathrm{Gal}(N/K)\) of \(N\) over the cyclotomic field \(K=\mathbb{Q}(ζ_5)\), by making use of theorems by Hasse and Iwasawa on the He…