Let I be a toric ideal. We say I is robust if its universal Gröbner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graphtheoretic conditions on the set of circuits of G.Theorem 1.2. I G is robust if and only if the following conditions are satisfied.R1: No circuit of G has an even chord, R2: No circuit of G has a bridge, R3: No circuit of G contains an effective crossing, and R4: No circuit of G shares exactly one edge (and no other vertices) with another circuit such that the shared edge is part of a cyclic block in both circuits.

We use Galois cohomology to study the p-rank of the class group of Q(N 1/p ), where N ≡ 1 mod p is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5, we prove a complete characterization of the 5-rank of the class group of Q(N 1/5 ) in terms of whether or not (N −1)/2 k=1 k k and √ 5−1 2 are 5th powers mod N .