We give the first polynomial-time algorithm for checking whether the Galois group Gal (f) of an input polynomial f(X) ∈ Q[X] is nilpotent: the running time of our algorithm is bounded by a polynomial in the size of the coefficients of f and the degree of f . Additionally, we give a deterministic polynomial-time algorithm that, when given as input a polynomial f(X) ∈ Q[X] with nilpotent Galois g...

For a finite dimensional monomial algebra Λ over a field K we show that the Hochschild cohomology ring of Λ modulo the ideal generated by homogeneous nilpotent elements is a commutative finitely generated Kalgebra of Krull dimension at most one. This was conjectured to be true for any finite dimensional algebra over a field in [13].

We announce methods for e cient management of solvable matrix groups over nite elds. We show that solvability and nilpotence can be tested in polynomial-time. Such e ciency seems unlikely for membership-testing, which subsumes the discrete-log problem. However, assuming that the primes in jGj (other than the eld characteristic) are polynomiallybounded, membership-testing and many other computat...

Note: These are my notes on part of chapter 6 of Ravenel’s Orange Book (Nilpotence and Periodicity in Stable Homotopy Theory), prepared for a talk in the graduate student topology seminar at Johns Hopkins. Jon Beardsley gave a talk on the first part of chapter 6, which reduces the proof of the Periodicity Theorem to constructing an example of a single type n spectrum with a vn-self map. For not...