Discriminant is a basic invariant associated with an algebraic number field. Its notion was
first introduced by Dedekind in 1871. The problem of effective computation of discriminant as
well as an integral basis of an infinite family of algebraic number fields which are defined over
the field $\mathbb{Q}$ of rational numbers by certain types of irreducible polynomials has been tackled

To a triple of a ring \(R\), an \(R\)-module \(M\), and an ideal \(I\) we can associate the local cohomology modules \(H^n_I(M)\). One classical problem dating back to a question of Grothendieck is to identify when these vanish for all \(n > i\) and all modules \(M\), the smallest such \(i\) we denote the cohomological dimension of \(R\) with respect to \(I\). Grothendieck showed that this is bounded by \(d = \dim R\), while a very simple condition on \(\widehat{R}\) and \(I\widehat{R}\) controls the vanishing at \(d\).

Given an ideal \(I\) in a regular local ring \(A\), the cohomological dimension of \(I\) in \(A\) is the index of the highest non-vanishing local cohomology of \(A\) supported at \(I\). Determining effective upper bounds on the cohomological dimension in terms of topological invariants of \(\textrm{Spec}(A/I)\) is a central problem in commutative algebra: foundational results include the Hartshorne--Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem.

Consider a finite subgroup of the general linear group, with its natural action on a polynomial ring.  We discuss how the local cohomology module of the invariant ring compares with the invariant part of the local cohomology of the polynomial ring.  This has various consequences, such as for the \(a\)-invariant and for the Hilbert series.  This is joint work with Kriti Goel and Jack Jeffries.