5 problems.
- Order of \(\GL_2(\FF_p)\) and its subgroups of order \(p\)
- Length of a module, composition series, and additivity of length along exact sequences
- The sum of group elements is central and nilpotent in \(\mathbb{F}[G]\) when \(\mathrm{char}\,\mathbb{F}=p\) and \(|G|=p^n\), so \(\mathbb{F}[G]\) is not semisimple
- \(\ZZ[x]/(f)\) is a finitely generated \(\ZZ\)-module if and only if the leading coefficient of \(f\) is \(\pm 1\)
- An algebraic extension of a characteristic-zero field in which every polynomial over \(F\) has a root is algebraically closed