I. M. Isaacs, David Petrie Moulton
The main result of this paper is that if E is a field extension of finite odd degree over a real field Q, and if E is a repeated radical extension of Q, then every intermediate field is also a repeated radical extension of Q. This paper also contains a number of other results about repeated radical extensions.
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On the degree of repeated radical extensions
本文回答了莫德尔1953年关于重复根式扩张的问题，给出了域扩张次数等于各根指数乘积的充要条件。
Let $L/K$ be a finite extension of congruence function fields. We say that $L/K$ is a {\it radical extension} if $L$ is generated by roots of polynomials $u^{M}-α\in K[u]$, where $u^{M}$ is the action of Carlitz-Hayes. We study a special class of these extensions, the {\it radical cyclotomic} extensions. We prove that any radical cyclotomic extension has order a power of the characteristic of $K$. We also give bounds for the Carlitz-Hayes torsion of these extensions.
Integers for Simple Radical Extensions
本文确定了简单根式扩张数域的整数环与判别式。
We show that finite Galois extensions with cyclic Galois group are radical.
For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L/F)$ is a power of $p$, by a Hecke's theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.
On the basis of a previous work, we elaborate a new description of the Kummer radical associated to the first layers of $\mathbb{Z}_\ell$--extensions of a number fields K, by using inverse limits for the norm maps in the cyclotomic $\mathbb{Z}_\ell$-extension. Our main result contains, as an obvious consequence, the inclusions provided by Soogil Seo in a set of papers. By the same way we also give in the last section a similar description of the Tate kernel for universal symbols in $K_2(K)$.
For a radical extension K of odd prime degree the ring O_K of integers is constructed as a product of subrings with the following property: for all prime divisors q of the discriminant of O_K there is a q-maximal factor. The discriminant of O_K is the greatest common divisor of the discriminants of all factors. The results are applied to give a criterion for the monogeneity of K where the opposite is not true.
We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).
We give necessary and sufficient conditions for the Kummer extension $K:=\mathbb{Q}\left(ζ_n,\sqrt[n]α\right)$ to be monogenic over $\mathbb{Q}(ζ_n)$ with $\sqrt[n]α$ as a generator, i.e., for $\mathcal{O}_K=\mathbb{Z}\left[ζ_n\right]\left[\sqrt[n]α\right]$. We generalize these ideas to radical extensions of an arbitrary number field $L$ and provide necessary and sufficient conditions for $\sqrt[n]α$ to generate a power $\mathcal{O}_L$-basis for $\mathcal{O}_{L\left(\sqrt[n]α\right)}$. We also give sufficient conditions for $K$ to be non-monogenic over $\mathbb{Q}$ and establish a general crit…
Small points in radical extensions of number fields
本文针对数域根式扩张中的小点问题，给出确保特定集合具有Bogomolov性质的通用判据，作为应用得到Rémond猜想的新案例，论证基于p-adic李扩张的相关结果。
Prime Splitting and Common Index Divisors in Radical Extensions
本文研究根式扩张中奇素数的显式分解，并据此刻画公共指数因子，证明任何造成幂整基局部障碍的素数 $p$ 都必整除根指数 $n$。
Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of Phi. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers O_K.
Let $ K $ be a number field and let $ L/K $ be a tamely ramified radical extension of prime degree $ p $. If $ K $ contains a primitive $ p^{th} $ root of unity then $ L/K $ is a cyclic Kummer extension; in this case the group algebra $ K[G] $ (with $ G=\mbox{Gal}(L/K) $) gives the unique Hopf-Galois structure on $ L/K $, the ring of algebraic integers $ \mathfrak{O}_L $ is locally free over $ \mathfrak{O}_{K}[G] $ by Noether's theorem, and Gómez Ayala has determined a criterion for $ \mathfrak{O}_L $ to be a free $ \mathfrak{O}_{K}[G] $-module. If $ K $ does not contain a primitive $ p^{th} $…
For an odd prime p and a number field F containing a primitive p-th root of unity, we describe the Kummer radical A_F of the first layers of all the Z_p-extensions of F in terms of universal norms of p-units along the cyclotomic tower of F . We also study "twisted" radicals related to AF .
Let $\mathbf{T}$ be a neutral Tannakian category over a field of characteristic zero with unit object $\mathbf{1}$, and equipped with a filtration $W_\cdot$ similar to the weight filtration on mixed motives. Let $M$ be an object of $\mathbf{T}$, and $\underline{\mathfrak{u}}(M)\subset W_{-1}\underline{Hom}(M,M)$ the Lie algebra of the kernel of the natural surjection from the fundamental group of $M$ to the fundamental group of $Gr^WM$. A result of Deligne gives a characterization of $\underline{\mathfrak{u}}(M)$ in terms of the extensions $0\longrightarrow W_pM \longrightarrow M \longrightarr…
Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$
本文明确描述了整数素数2在根式扩张$\mathbb{Q}(\sqrt[n]{a})$中的分裂，结合已有工作完整刻画了任意素数的分裂，并分类了公共指数因子，进而推广到公共$N$-指数因子，还构造了无非公共指数因子的非单生成域及需要$N+1$个环生成元的数环。
On blended extensions in filtered abelian categories and motives with maximal unipotent radicals
本文推广 Grothendieck 混合扩张理论至任意有限步滤过，用于分类具有固定相伴分次的对象，并应用于混合 motive，在分次独立条件下证明同调分类结果。
Free Pro-2 Product Decompositions and Hilbert's Theorem 90 for the Kaplansky Radical
本文针对有限平方类域建立了确保Kaplansky根的Hilbert第90型性质的伽罗瓦理论判据，将其应用于初等类型极大pro-2伽罗瓦群的域等，为有限维H猜想的已知正例提供了统一解释。
Combinatorial Extensions on Andrews and El Bachraoui's Almost-Distinct Partitions
针对Andrews和El Bachraoui提出的最小部分重复、其他部分不同的整数分拆相关恒等式，本文给出其无限族的组合证明，还研究仅最大部分可重复的类似分拆，得到不同部分分拆数的无限族不等式。
A generalization of Kummer theory to Hopf-Galois extensions
本文通过引入$H$-Kummer扩张的概念，将经典Kummer理论推广至Hopf-Galois扩张，将其刻画为与分圆扩张线性无关的根式扩张，并利用乘积Hopf-Galois结构推导了数域与$p$-adic域上整数环自由性的判定准则。
If K_0 is the fraction field of the Witt vectors over an algebraically closed field k of characteristic p, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions K_0(zeta_{p^r}, sqrt[p^r]{a})/K_0, where a is in K_0(zeta_{p^r}). Here zeta_{p^r} is a primitive p^r-th root of unity. In certain cases, including when a is in K_0 and p=2, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of Q obtained by adjoining roots of unity and radicals.
A cohomological translation of the Kaplansky radical for profinite groups
引入任意profinite群和素数p的Kaplansky根的上同调类比，证明对于域，群论版本的H-猜想等价于原猜想，并建立其对于多种域类的成立性。
I provide a systematic construction of points, defined over finite radical extensions, on any Legendre curve over any field of characteristic not equal two. This includes as special case Douglas Ulmer's construction of rational points over a rational function field in characteristic $p>0$. In particular I show that if $n\geq 4$ is any even integer and not divisible by the characteristic of the field then any elliptic curve $E$ over this field has at least $2n$ rational points over a finite solvable field extension. Under additional hypothesis, when the ground field is a number field, I show th…
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