## What is algebraic number theory?

This is quite a difficult question to answer precisely. This branch of maths has evolved so much over the last $150$ or so years.

On a basic level it is two things. On the one hand it is essentially the study of number theory by using algebraic methods (groups, rings, fields, modules etc). On the other hand it is essentially the study of algebraic numbers (numbers that are roots of polynomials over certain fields).

It has become so much more than this.

The origins of algebraic number theory really came from mathematicians wanting to solve Diophantine equations.

For example, if we wanted to solve the equation $y^2 + 2 = x^3$ in integers, we could only get so far using elementary methods (or would have to really make a big leap somewhere in the proof). We would probably get as far as seeing that $y$ must be odd (since if it were even then the left hand side would have to be divisible by $8$).

However, an algebraic number theorist would not be able to resist the urge to factorise as:

$(y + \sqrt{-2})(y - \sqrt{-2}) = x^3$

and work in the ring $\mathbb{Z}[\sqrt{-2}] = \{a+b\sqrt{-2} | a,b\in\mathbb{Z}\}$.

It turns out that this ring behaves a lot like the integers in some sense. We definately have unique factorisation into irreducible elements (upto ordering and multiplication by units).

Compare this with the story in $\mathbb{Z}$; each non-zero, non-unit integer has a unique expression (upto ordering) as a product of prime numbers and a unit (the units here being $\pm 1$).

This fact about $\mathbb{Z}[\sqrt{-2}]$ would have been assumed by mathematicians in Euler’s day but as we will see, sometimes unique factorisation doesn’t hold in rings like these.

Using this uniqueness of factorisation we would be able to progress blindly just as if we were working in $\mathbb{Z}$, since it is easily shown that when $y$ is odd, the two elements $(y+\sqrt{-2})$ and $(y-\sqrt{-2})$ are “coprime” in  $\mathbb{Z}[\sqrt{-2}]$.

So by unique factorisation we would have to have that $(y + \sqrt{-2}) = u(a + b\sqrt{-2})^3$ for some unit $u$ of $\mathbb{Z}[\sqrt{-2}]$ and $a,b\in\mathbb{Z}$ (again compare with $\mathbb{Z}$, if two coprime integers multiply to give a cube, then each must be a cube).

Since the only units of $\mathbb{Z}[\sqrt{-2}]$ are $\pm 1$ we may assume that $u=1$ (both units are cubes and so it does not matter which one we take).

From here we find it easy, since equating $\sqrt{-2}$ terms gives us an integer equation $1 = b(3a^2 - 2b^2)$. This equation only has two solutions $(a,b) = (\pm 1, 1)$, leading to the solutions $(x,y) = (3, \pm 5)$.

So not only does this equation only have finitely many solutions in integers, but we see the clear advantages of working in “bigger” rings.

A specific famous Diophantine that spurred on the foundations of algebraic number theory was the one given in Fermat’s Last Theorem, $x^n + y^n = z^n$ for $n > 2$

As is commonly known, Fermat conjectured that this equation had no solutions in non-zero integers.

The case $n=4$ was proved by Fermat himself (via descent), leaving the cases for odd prime $n$ to be solved (having each of these cases will provide a full solution).

Lamé tried to solve the odd case by introducing a primitive $n$th root of unity $\zeta = e^{\frac{2\pi i}{n}}$ and factorising the equation as:

$(x+y)(x+\zeta y)(x + \zeta^2 y)\mathellipsis(x + \zeta^{n-1}y) = z^n$.

Here we now extend to work in the ring:

$\mathbb{Z}[\zeta]$ = $\{a_0 + a_1 {\zeta} + a_2 {\zeta}^2 + \mathellipsis$ $+\, a_{n-2} {\zeta}^{n-2}$ $| a_0 , a_1 , a_2 ,\mathellipsis a_{n-2}\in\mathbb{Z}\}$

He assumed unique factorisation into irreducibles as above to claim that each term on the left would have to be the product of a unit and an $n$th power in $\mathbb{Z}[\zeta]$, deriving a contradiction (hence “proving” Fermat’s Last Theorem).

Gauss also used such rings to look at higher reciprocity laws, generalising quadratic reciprocity but we shall leave that for another time.

Lamé’s argument was false since unique factorisation into irreducibles does not always work in these rings (Kummer provided a counter-example for the case $n=23$).

So it then became a mission to classify when we do have unique factorisation and when we don’t. Also, when we don’t is there anything we can do to restore it?

These are the main tasks of algebraic number theory (well classically). Modern algebraic number theory goes beyond these questions but to investigate modern breakthroughs we will have to first consider classical ones.

Suffice to say here that Dedekind was the saviour of the day. He provided the key breakthrough for unique factorisation, that we should look at the ideals of such rings rather than the elements.

(Well actually Kummer was the first to look into this by constructing “ideal numbers”, certain numbers that lie in bigger rings, but Dedekind was the first to make this more algebraic also without having to leave the original ring).

Of course, we need to specify what rings we will be working with and this will be the topic of the next post, an introduction to the basic notions in algebraic number theory.

I personally love number theory and find algebraic number theory really interesting due to the amazing blend of concrete examples, applications and abstractness. I hope to convey this love to others in this blog (along with other good stuff I manage to find).