4 problems.
- A finitely generated group has finitely many subgroups of index \(n\), and the possible numbers of index-\(p\) subgroups of a finitely generated abelian group
- A group of order \(2013=3\cdot 11\cdot 61\) has a cyclic normal subgroup of index \(3\), and \(11\) divides \(|Z(G)|\)
- Unique Galois quartic \(\QQ(\sqrt{d_1},\sqrt{d_2})\) in a nonabelian Galois octic \(K/\QQ\); \(d_1,d_2>0\) when \(G=Q_8\)
- \((5,x^2+2)\) is prime in \(\mathbb{Z}[x,y]\) with infinitely many maximal extensions