Generalized Steinberg relations for absolute Galois groups.

Ido Efrat Ben Gurion University
The absolute Galois group $G_F$ of a field $F$ describes the symmetry patterns among the roots of all polynomials over $F$. While its group-theoretic structure is in general still a mystery, its cohomology is completely described in elementary terms by the seminal work of Voevodsky and Rost. Specifically, if $F$ contains a root of unity of order $n$, then the cohomology algebra $H^\bullet(G_F,\mathbb{Z}/n)$ is generated by the multiplicative group of $F$ with a very simple set of defining relations, called the Steinberg relations. In recent years there has been considerable interest in higher cohomological operations for $G_F$ - such as Massey products. We will show how the classical Steinberg relations extend to this context.

Hilbertian pseudo-algebraically closed fields: Origins and applications

Arno Fehm TU Dresden
The first topic of this talk are a class of fields that first emerged in the work of Ax on the model theory of finite fields, and for which Frey suggested the name pseudo-algebraically closed (PAC). In the 60's and 70's, Jarden pioneered both the algebraic and the model theoretic study of these fields. The term Hilbertian was coined by Lang and derives from Hilbert's irreducibility theorem. Also here Jarden made seminal contributions, and the book Field Arithmetic that he coauthored remains the standard reference. I will review some of Jarden's contributions to these subjects and then present two recent applications of the theory of Hilbertian PAC fields to other areas: Inverse Galois Theory, and Hilbert's Tenth Problem on the decidability of diophantine equations.

On the minimal ramification problem for the symmetric and alternating groups

Alexei Entin Tel Aviv University
The minimal ramification problem asks for any given finite group G what is the minimal number r(G) of primes (including infinity) that ramify in some extension of Q with Galois group G. In the case of the groups S_n and A_n, a conjecture of Boston and Markin predicts that r(S_n)=r(A_n)=1. While this is still open, progress was made by Bary-Soroker and Schlank who were able to show that r(S_n)<=4 for all n. This result relies on the Green-Tao-Ziegler theorem on solutions in primes to linear systems of equations. We will discuss further progress on this problem, showing that r(S_n)<=3 for n odd and that r(A_n) is bounded provided n=0,1(mod 4). Key ingredients in these results include the specialization method of Bary-Soroker and Schlank and recent advances on simultaneous prime values of a quadratic and linear form due to Lam, Schindler and Xiao. In the talk I will give a survey of the minimal ramification problem and the Boston-Markin conjecture, discuss previous work on the subject (with emphasis on the work of Bary-Soroker and Schlank), state the new results and discuss the main ingredients of the proofs.