### Algebra SeminarCounting Towers of Number FieldsBrandon Alberts (University of Connecticut)

Wednesday, September 25, 2019
11:15am – 12:05pm

Storrs Campus
Monteith 313

Fix a number field $$K$$ and a finite transitive subgroup $$G \le S_n$$. Malle's conjecture proposes asymptotics for counting the number of $$G$$-extensions of number fields $$F/K$$ with discriminant bounded above by $$X$$. A recent and fruitful approach to this problem introduced by Lemke Oliver, Wang, and Wood is to count inductively. Fix a normal subgroup $$T$$ of $$G$$. Step one: for each $$G/T$$-extension $$L/K$$, first count the number of towers of fields $$F/L/K$$ with $${\rm Gal}(F/L) \cong T$$ and $${\rm Gal}(F/K)\cong G$$ with discriminant bounded above by $$X$$. Step two: sum over all choices for the $$G/T$$-extension $$L/K$$. In this talk we discuss the close relationship between step one of this method and the first Galois cohomology group. This approach suggests a refinement of Malle's conjecture which gives new insight into the problem. We give the solution to step one when $$T$$ is an abelian normal subgroup of $$G$$, and convert this into nontrivial lower bounds for Malle's conjecture whenever $$G$$ has an abelian normal subgroup.

Contact:

Mihai Fulger, mihai.fulger@uconn.edu

Algebra Seminar (primary), UConn Master Calendar