8 problems.
- A finite group is abelian if conjugacy class representatives pairwise commute
- Groups of order \(105\): normal Sylow \(5\)- and \(7\)-subgroups, a cyclic subgroup of order \(35\), and cyclicity when the Sylow \(3\)-subgroup is normal
- A ring of idempotents has characteristic \(2\) and is commutative
- The degree of \(\FF_q(\omega)\) is the multiplicative order of \(q\) modulo \(n\)
- Torsion elements form a submodule over an integral domain but not in general, and every module over a ring with zero-divisors has torsion
- The nilradical is contained in every prime, primes avoiding powers of a non-nilpotent, and rings with a unique prime consist of nilpotents and units
- \(\mathbb{Q}(\zeta_n)/\mathbb{Q}\) is Galois with group \((\mathbb{Z}/n\mathbb{Z})^\times\), and the subfields of \(\mathbb{Q}(\zeta_{20})\)
- The dual of an integer lattice: \(\Lambda\subset\Lambda^\vee\), \(\det M\neq 0\), and \(|\Lambda^\vee/\Lambda|=\det M\)