Expository problems

Question

In one article on his blog; T. Gowers talks about ''expository problems''.

For those too lazy to click that link, a quote of a quote:

Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves.

I'm sure that all of us can value good solutions to expository problems, and that all of us were sometimes really frustrated by some expository problems.

So, what expository problems do you know? What's that one thing in math you could never intuitively grasp? If you are a member of faculty, have you ever noticed some topics in your field which are particularly hard to get for your students (and you don't really get why)?

P.S. I'm sorry if this question is too ''soft'' for math.stackexchange, but I think it's really interesting.

Answer 1

I never understood intuitively why pi is irrational.

Answer 2

Here are two statements:

$\bullet$ The ring $\mathbb Z/4 \mathbb Z$ has 4 elements but is not a field.
$\bullet \bullet$ There exists a field with 4 elements.

I have noticed year after year that it is quite difficult for students to believe that both can be true simultaneously.

The remedy is to describe the field very explicitly as $$\mathbb F_4=\frac {\mathbb F_2[X]}{\langle X^2+X+1\rangle}=\lbrace 0,1,x,1+x \rbrace $$

The expository problem might be to make crystal-clear what these cabalistic equalities mean and why this $\mathbb F_4$ is indeed a field with four elements.
And then to explain why the use of $\mathbb F_2$ and $X^2+X+1$ on the road leading to $\mathbb F_4$ is the kernel of the "bootstrapping" method , which allows us to construct a big finite field $K$ from a small finite field $k$ by taking an irreducible polynomial $f(X)\in k[X]$ and building the quotient $$K=k[X]/\langle f(X)\rangle$$.