Algebraic number theory topics for undergrads

Question

What are some interesting topics or problems in algebraic number theory which could be presented to students in a first undergraduate algebra course (which covers some elementary number theory, groups, and rings)?

Answer 1

Sounds very much like our first course in algebraic structures. Examples that might or might not fit the bill (I have only tried a few of these, but will get a chance to do so soon):

1. Pell equations. May be while looking for units of the ring $\Bbb{Z}[\sqrt2]$? Start by observing that $u=1+\sqrt2$ is a unit because $(1+\sqrt2)(1-\sqrt2)=-1$. Because the units of a ring form a group, its powers $$u^n=(1+\sqrt2)^n=a_n+b_n\sqrt2$$ will also be units. Because $\sqrt2\mapsto-\sqrt2$ extends to a ring automorphism, the inverse is $(-1)^n(a_n-b_n\sqrt2)$. This also yields a sequence of good rational approximations to $\sqrt2$, so one can build some purely computational exercises out of this as well (in my experience the weaker students appreciate those). The fact that the unit group is $C_\infty\times C_2$ must wait :-)
2. Fields of size $p^2$ as quotients of rings of integers of number fields. $\Bbb{Z}[i]/\langle p\rangle$ is such a field when $p=3$ (compute the inverses), but with $p=5$ its is not. Why? May try quotients of other rings as well?
3. If you cover fields of fractions, you can also make basic exercises about the rings of rational $p$-adic integers. Interesting subrings of $\Bbb{Q}$?

This is a bit of a stretch - sorry about that.

Answer 2

I am reading this delightful paper by Harald Helfgott Growth and Generation in $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ where he explores the Cayley graphs whose vertices are $2\times 2$ invertible matrices with entries in the finite fiele $\mathbb{F}_p$. Two edges are connected if they are joined by elements of a generating set.

$$ A = \left\{ \left(\begin{array}{cc} 1 & 1 \\ 0 & 1\end{array}\right) , \left(\begin{array}{cc} 1 & 0 \\ 1 & 1\end{array}\right) \right\}$$

He asks for the diameter of this graph - the length of the longest path between two elements. The result was known using difficult spectral methods for the above generating set and unknown for the generating set below. Basically change 1 into 3.

$$ A = \left\{ \left(\begin{array}{cc} 1 & 3 \\ 0 & 1\end{array}\right) , \left(\begin{array}{cc} 1 & 0 \\ 3 & 1\end{array}\right) \right\}$$

This would be a great chance for to explore the interplay between groups and generators, graphs and eigenvalues.

The full results of the paper are certainly out of reach for an intro algebra class, however this is for the better. The statement of the problem should be enough; partial result lead to meaningful exploration on a modern research-type problem.

Answer 3

I would like to see:

• Fermat's Last Theorem
• Quadratic Residues
• Quasigroups

, all of which have undergone major transformations recently, and offer some form of 'closure' (my word - they all have results, not just conjectures).