6 problems.
- Groups of order 2009 and their intermediate subgroups
- The valuation ring of a discrete valuation, and the \(p\)-adic valuation on \(\mathbb{Q}\)
- Quadratic extensions are \(F(\sqrt{D})\) when \(\operatorname{char} F\neq 2\), and \([F(\sqrt{D_1},\sqrt{D_2}):F]\)
- A root extension of an irreducible polynomial: existence, basis, and \(\theta^{-1}\); irreducibility of \(x^3+9x+6\) and the inverse of \(1+\theta\)
- Chain conditions force injective or surjective module endomorphisms to be automorphisms
- Schur’s lemma: \(\operatorname{Hom}_{kG}(V,V)=k\) for an irreducible representation over an algebraically closed field