Here is a list of some questions from Math StackExchange. These are not particularly tough questions but ones I find nice. (And want to keep them here for quick reference.)
If \(K/\mathbb{Q}\) is finite, then there exists a finite extension \(E/K\) which is not normal.
An elementary way of counting the number of generators of \(\mathbb{F}_{q^{n}}\) over \(\mathbb{F}_{q}\).
Interestingly, a similar question came up in a test I gave few months later. I started doing it in a more complicated way (similar to my other answer on that question) but then I remembered I had done this special case in an easier way on this post.
Computing the Galois group of \(\mathbb{Q}(\sqrt[6]{-3})/\mathbb{Q}\).
Is the subgroup of divisible elements a divisible a subgroup?
At the time of writing this, this is probably the answer I consider to be my best one.
A group of order \(pq\) is abelian when \(p \leq q\) and \(p \nmid q-1\).
Is \(\arctan(2)\) a rational multiple of \(\pi\)?
While the question doesn’t look like it belongs in this tag, my answer solves it using unique factorisation in the domain \(\mathbb{Z}[\sqrt{-1}]\).
Proving that \((x^2 + 2)^n + 5(x^{2n-1} + 10x^2 + 5)\) is irreducible
I like this question because it introduced me to a new irreduciblity criterion.
Proving that a certain composition of two integer polynomials has no integer solution.
\((x^{2} - 2)(x^{2} - 3)(x^{2} - 6)\) has a root in \(\mathbb{F}_{p}\) for every prime \(p\).