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.)

My answers

galois-theory

1. If \(K/\mathbb{Q}\) is finite, then there exists a finite extension \(E/K\) which is not normal.

2. 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.

3. Computing the Galois group of \(\mathbb{Q}(\sqrt[6]{-3})/\mathbb{Q}\).

group-theory

1. 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.

2. A group of order \(pq\) is abelian when \(p \leq q\) and \(p \nmid q-1\).

3. There is no simple group of order \(2^73^2\).

4. \(A\cap C = 1\) if \(A\) is a non-abelian normal simple subgroup and \(C\) is the centralizer of \(A\).

5. Equivalence of wedge products.

6. If a group has subgroups of orders \(r\) and \(s\), then it has a subgroup of order \(rs\) when \(r\) and \(s\) are coprime.

7. Every proper subgroup of \(\mathbb{Z}(p^{\infty})\) is of the form \(\left\langle \overline{\dfrac{1}{p^n}} \right\rangle\) for some \(n>0\).

commutative-algebra

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 \(\mathbb{Z}[\sqrt{-1}]\).

2. 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.

analysis

1. Let \(g \in C^1(\mathbb{R^2}, \mathbb{R})\). Show that an injective \(f \in C^1 ((-1,1), \mathbb{R^2})\) exists, so \(g \circ f\) is constant.

2. If the limit \(\displaystyle\lim_{t\to \infty}f(t)\) exists and \(f'\) is uniformly continuous, then \(\displaystyle\lim_{t\to \infty}f'(t) = 0\).

3. \(\displaystyle\lim_{x\to 1^{+}}\zeta(x) = \infty\).

4. \(\displaystyle\lim_{n\to \infty}\left(1 + \dfrac{z}{n}\right)^n\) does not converge uniformly on \(\Bbb C\).

5. \(e^{-\cos^2x}\) has a unique fixed point.

6. How many decimal expansions can a real number have?

7. If \(f\) is analytic on \(\{\lvert z \rvert \leq 1\}\), and \(f\) is real on \(\{\lvert z \rvert = 1\}\), then \(f\) is a constant.

topology

1. \(S^3 \setminus S^1\) is connected.

2. \(\mathbb{R}/{\sim}\) is compact, where \(x \sim y\) iff \(x = 2^{n}y\).

number-theory

1. Proving that a certain composition of two integer polynomials has no integer solution.

2. \(23a^2\) is never the sum of \(3\) squares.

3. Finding all \(t\) such that \(\varphi(t) \mid t\).

4. \((x^{2} - 2)(x^{2} - 3)(x^{2} - 6)\) has a root in \(\mathbb{F}_{p}\) for every prime \(p\).

linear-algebra

1. Let \(V\) be a finite-dimensional vector space and \(W,W_1,W_2\) subspaces of \(V\), such that \(V=W_1\oplus W_2\), \(W \cap W_2=\{0\}\) and \(\dim W= \dim W_1\). Prove that there exists a linear transformation \(f:W_1\to W_2\), such that \(W=\{v\in V \mid \exists w_1\in W_1:v=w_1+f(w_1)\}\).

2. Do there exist vector spaces without an inner product?

3. A question about vector spaces over arbitrary fields.

Other questions

1. A weak cancellation property of monoids does not imply cancellation.

2. Covering of a compact set by finitely many balls of the form \(B(x_i, 2r)\) such that the balls \(B(x_i, r)\)s are disjoint.

3. Any infinite sequence of functions \(g_n:\Bbb R\to\Bbb R\) can be written as composition of functions from a finite set.

4. \(f(0)=f(1)=0\), \(f(x)=\dfrac{f(x+h)+f(x-h)}{2}\) implies \(f(x)=0\) for \(x \in [0, 1]\). (\(f\) is continuous.)

5. Existence of a continuous function \(f\) such that it takes the value \(0\) on a set with positive measure but is not locally constant at any such point.

6. Are the arctangents of 1, 1.5, 2 linearly independent over the rationals?