5 problems.
- An irreducible quintic over \(\mathbb{Q}\) with three real roots is not solvable by radicals, and a quadratic Galois subfield of its splitting field
- A finite field has irreducible polynomials of every positive degree
- Two-generated modules over a PID are \(R/(a)\oplus R/(b)\), and when two such sums are isomorphic
- The subgroup of \(\GL_3(\CC)\) generated by permutation matrices of \(S_3\) and \(\mathrm{diag}(i,1,1)\)
- Irreducible \(\CC G\)-modules have dimension at most \(\sqrt{|G|}\), with an example of dimension \(\lfloor\sqrt{|G|}\rfloor\)