how to split a separable algebra?

Question

I'm trying to factor ideals in a function field (more precisely, ideals in a maximal order of a function field), and I've come across a step in the published Buchman-Lenstra algorithm which works in prime characteristic, but I'm trying to figure how to do it in characteristic zero.

We start a Dedekind domain $O$, given to us as an $R$-algebra using a basis in some polynomial ring, and $I$, a maximal ideal of $R$. The goal is to factor $I$'s extension in $O$, $I^e$. We use basic ideal operations (addition, division, powering) to perform a square-free decomposition on $I^e$'s radical. Now we've reduced the problem to factoring an ideal that is the product of unique maximal ideals, and this implies that the quotient $O/I^e$ is a separable (étale) algebra over the field $K=R/I$.

We know that such an algebra can be expressed as a direct product of fields (cf. Chinese remainder theorem); the problem is to actually express it that way.

The published algorithms that I've found ([Coh93] Sections 6.2.2 and 6.2.4; [St12] Algorithms 4.3.7 and 4.3.9) work over a finite field, say $F_p$, and make use of the fact that the kernel of $x\to x^p-x$ is $F_p$ (cf. Frobenius map), so the dimension of the kernel of this map, applied to the algebra, will be the number of fields in the direct product. A few steps later we find sub-algebras that product together to form the original algebra.

Obviously, this depends on a property of finite fields, so it works fine for number fields (the ideals we're factoring are themselves generated by prime numbers) and global function fields (the base field is a finite field), but I'm interested in function fields over $\mathbb{Q}$ and $\overline{\mathbb{Q}}$ where the ideals are like $(x)$ and $(x-1)$. My questions are:

1. how can such an algebra be split in characteristic zero? and

2. is this algorithm published somewhere?

I can think of one way to do it myself. We already have a basis for the algebra, so construct matrices to express all of its multiplication-by-basis-element operations, then put them in a canonical form for simultaneous similarity. I haven't worked out all of the details, so this may or may not work.

I know that programs like KASH and MAGMA can factor these ideals, but I haven't been able to find any references to their algorithms. Maybe somebody can help?

Answer 1

I found a paper that answers my question. It's How to Compute the Wedderburn Decomposition of a Finite-Dimensional Associative Algebra by Murray R. Bremner. From the abstract:

This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field.

Wedderburn showed that every finite-dimensional semisimple algebra can be expressed as the direct sum of simple algebras, and since commutative simple algebras are fields, the Wedderburn decomposition is what I'm looking for.