Trace forms of Galois extensions in the presence of a fourth root of unity

Quotients of absolute Galois groups which determine the entire Galois cohomology

For prime power $q=p^d$ and a field $F$ containing a root of unity of order $q$ we show that the Galois cohomology ring $H^*(G_F,\dbZ/q)$ is determined by a quotient $G_F^{[3]}$ of the absolute Galois group $G_F$ related to its descending $q$-central sequence. Conversely, we show that $G_F^{[3]}$ is determined by the lower cohomology of $G_F$. This is used to give new examples of pro-$p$ groups which do not occur as absolute Galois groups of fields.

Finite quotients of Galois pro-$p$ groups and rigid fields

For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.

Iterative q difference Galois Theory

We propose in this paper a Galois theory of $q$-difference equations where q is a root of unity. This theory is the q difference analogue of the Galois theory of iterative differential equations, that is differential equations over fields of positive characteristic. This theory contains and generalizes the Galois theory of q difference equations developed by Singer and van der Put.

An elementary computation of the Galois groups of symmetric sextic trinomials

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases: whether or not the cubic roots of unity belong to $K$. We also give a criterion to determine whether a polynomial as $f(x)$ is irreducible, when $F$ is a finite field. Moreover, at the end of the paper we also give a complete list of all the possible subfields of $F$.

Galois module structure of $p^{\text{th}}$ power classes of abelian extensions of local fields

In this paper, we describe the Galois module structure of $J=\mathbf{K}^{\times}/\mathbf{K}^{\times p}$, where $\mathbf{K}$ is an extension of a local field $\mathbf{k}$ containing a primitive $p$-th root of unity: for instance, if $\mathbf{K}/\mathbf{k}$ is a $p$-elementary abelian extension, we prove that $J$ is a module of constant Jordan type, with stable Jordan type $[1]^2$, which, in a way, extends the result of J. Mináč and J. Swallow. Also, we take profit from our proof by computing some invariants, which were previously introduced by A. Adem, W. Gao, D. B. Karageuzian and J. Mináč onl…

具有周期临界点的二次有理函数的伽罗瓦理论

Galois theory of quadratic rational functions with periodic critical points

针对二次有理函数的伽罗瓦树表示，本文通过研究迭代单值群，展示了临界点周期的二次有理函数的树伽罗瓦群非稠密稳定，否定相关问题并证明特定常数字段猜想。