Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

2 Answers 2

I never understood intuitively why pi is irrational.

share|improve this answer

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 $$K=k[X]/\langle f(X)\rangle$$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.