Some Galois extensions of quadratic extensions associated with Witt rings

二次PCF多项式绝对伽罗华群的定居元素

Settled Elements in Arboreal Galois Groups of Quadratic PCF Polynomials

研究二次PCF多项式迭代单子群的定居性质，证明其在数域中存在无限多个参数使对应的树形伽罗华表示密集定居。

Field theory and the Cohomology of Some Galois Groups

We prove that two arithmetically significant extensions of a field F coincide if and only if the Witt ring WF is a group ring Z/n[G]. Furthermore, working modulo squares with Galois groups which are 2-groups, we establish a theorem analogous to Hilbert's Theorem 90 and show that an identity linking the cohomological dimension of the Galois group of the quadratic closure of F, the length of a filtration on a certain module over a Galois group, and the dimension over Z/2 of the square class group of the field holds for a number of interesting families of fields. Finally we discuss the cohomology…

Arason's filtration of the Witt group of dyadic valued fields

Generalizing a theorem of Springer, we construct an extended Arason filtration by subgroups for the Witt group of quadratic forms of a general valued field, relating these subgroups with Witt-like groups of the residue field, in arbitrary characteristic. These Witt-like groups involve totally singular quadratic forms. In the case of a discretely valued field, we recover the original Arason filtration.

Two families of pro-p groups that are not absolute Galois groups

Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these families one has one-relator pro-$p$ groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology.

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。

One relator maximal pro-p Galois groups and the Koszulity conjectures

Let $p$ be a prime number and let ${K}$ be a field containing a root of 1 of order $p$. If the absolute Galois group $G_{K}$ satisfies $\dim H^1(G_{K},\mathbb{F}_p)<\infty$ and $\dim H^2(G_{K},\mathbb{F}_p)=1$, we show that L.~Positselski's and T.~Weigel's Koszulity conjectures are true for ${K}$. Also, under the above hypothesis we show that the $\mathbb{F}_p$-cohomology algebra of $G_{K}$ is the quadratic dual of the graded algebra $\mathrm{gr}_\bullet\mathbb{F}_p[G_{K}]$, induced by the powers of the augmentation ideal of the group algebra $\mathbb{F}_p[G_{K}]$, and these two algebras decom…