8 problems.
- \(p\)-groups have nontrivial center, groups of order \(p^2\) and \(5^2\cdot 7^2\) are abelian, and the isomorphism classes of groups of order \(5^2\cdot 7^2\)
- Splitting field, Galois group, and intermediate fields of \(x^4-4x^2+2\) over \(\QQ\)
- Galois groups of the compositum and intersection of two splitting fields over \(\QQ\)
- Jordan form of \(\begin{pmatrix}0&1&-2\\1&1&-3\\1&2&-4\end{pmatrix}\) and a conjugating matrix
- If \(b\) and \(x\) are units, a \(2\times 2\) product over a commutative ring that vanishes off the bottom-right corner is zero
- Aligned bases for the even-sum submodule \(M\subseteq\ZZ^4\) and the submodule \(N\) defined by \(4\mid(w-x)\), \(4\mid(x-y)\), \(4\mid(y-z)\), and the structure of \(M/N\)
- Invertibility outside \(\langle p\rangle\) and purity in \(\langle p\rangle\)-primary modules over a PID
- Maximal ideals of \(C[0,1]\) are \(\{f:f(x_0)=0\}\)