Ramification

Books

I tidied my mathematics books collection up a lot. It's embarassing to have a lot of unread/unstudied books.. but I organized them to help me find ones to study. I worked through a bit of Murtys Trancendental numbers book but some of the early proofs lacked enough explanation so I'm hesitant to study more of it in case I can't understand it well.

Algebraic number fields

When you move to algebraic number fields some primes split <https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions>, some don't and some "ramify". to ramify means it didn't split into unique factors (there was a repeated power).

• Prime ideals have unique factorization (proved using fractional ideals)
• Finitely many primes ramify, relation to discriminant
• The discriminant is never 1 (Geometry of numbers)
• https://en.wikipedia.org/wiki/Chebotarev's_density_theorem

More stuff

• Dirichlets units theorem
• Failure of unique factorization, finiteness of the class number
• The endomorphism algebra of an elliptic curve is a number field (complex multiplication)

The notes by wstein look good <http://wstein.org/129-05/notes/129.pdf>

I had a look at terry taos blog a bit and looked through the tags https://terrytao.wordpress.com/>. It might be good to work on sieve theory more: https://terrytao.wordpress.com/tag/sieve-theory/

Get on with it

I read over a nice proof that the ring of integers really is a ring using symmetric polynomials. That was new to me.

I'd like to understand first the theorem that finitely many primes ramify in an extension, and in particular they are exactly the primes from the discriminant. I looked at JS Milnes notes but it was confusing, I'm not familiar enough with discriminant.

Rough notes

The trace-pairing is similar to the inner product: <http://math.stackexchange.com/a/402164>

Ramification is equivalent to residue field being smaller than the full degree you would expect: In the case of the degree 2 extension Z[i], for any prime p the ring $Z[i]/(p)$ has size $\mathbb F_{p^2}$, except the ramified prime $2$ for which the field is $\mathbb F_{p}$.

A prime can ramify into any kind of form as long as it has a power greater than 1. In the Galois case things are a lot simpler (than the general case), the ramified primes will be perfect powers <https://en.wikipedia.org/wiki/Splitting_of_prime_ideals_in_Galois_extensions>. It was explained: "all the primes above a given prime are conjugate, so they have the same ramification index and inertia degree."

What I've heard is that the proof can be done in two ways: globally with direct calculations of trace and det like JS Milne does and then locally (Is Cassels and Frolich) by thinking about the completions at prime places. My intention for the new few days is to understand both proofs.