Algebraic Number Theory...

Im suprprised how huge even the early parts of this topic are. I'm learning a lot but really too much - there are so many concepts that I'm having a hard time organizing it all into a clear progression. Much of the theory is "high level" too in that simple statements require a lot of details to fill in - I have to be careful not to skip over and miss anything subtle.

ideals generalize to fractional ideals via modules. ideal norms. complex modular arithmetic systems, their sizes, dimensions. discrete valuation rings, completions, p-adic like systems. uniformizers.

Group of ideals

Probably the first important result is the group: Id(O_K).

The fractional ideals were introduced to prove unique factorization of an ideal into prime ideals. The set of fractional ideals is itself a group: freely generated by the prime ideals. That means each p^i is different - it's the group we would expect, no wrap-around or modulus.

We also have the valuation of an ideal with respect to some prime ideal: $v_p(I)$. And the formula $I = \prod_p p^(v_p(I))$.

$v_p(I)$ is defined as the minimum valuation of elements in the ideal: $\inf_{x \in I} v_p(x)$.

Cassels & Frolich also shows that the ideal group is isomorphic to the finite-support product of the ideal groups of the completions of discrete valuation rings at each prime. I think this is a kind of local-global principle.

My goal was to understand the primes that ramify, all of this is important background for that but there is a lot of very hard work in this already.

Ideals are lattices

A prime ideal $p$ might split into a product of powers of prime ideals - suppose $q$ is one of them. So $q$ divides $p$. In that case $q$ as a set contains $p$. In fact $p = q \int \mathbb Z$. As lattices, $p$ is a one dimension lattice on the number line - $q$ would be a higher dimensional lattice that agrees with $p$ on the line but has many more points outside of that.

That ideals are lattices suggest an invariant; the fundamental domain of the lattice. Hopefully this makes the notion of discriminant a bit less magical.

I drew some of the lattices out for primes that primes that ramify, are inert and that split in $\mathbb Z[i]$. It's very useful to do actual examples.

Splitting formula

There is a very nice formula about how prime ideals split when you move to an extension (of the ring of integers). $d = \sum_i^g e_i f_i$. In the galois case (because the galois group acts transitively on the ideals) this collapses down to $d = efg$.

This result comes from considering the dimension of modular arithmetic fields and using chinese remainder theorem. I was mostly able to prove it but there are some gaps I need to consult my books to fill in.

Notes

I'm just trying to get all the pieces together and prove them formally. There is so much machinary and tools needed so it's taking a long time.

The particular theorem I'm doing now is a key part of the $efg$ result. It is that $R/p = p^n/p^{n+1}$.

the proof in cassels only works for principal ideals but provides good intuition for the general case. It's tricker for non-principal <http://www1.spms.ntu.edu.sg/~frederique/ANT10.pdf> deals with it.

An interesting cubic diophantine equation $x^3 - 11y^3$ is solved in <http://people.math.gatech.edu/~mbaker/pdf/ANTBook.pdf>.