Maximal Orders and Decomposition II
Splitting Separable Algebras over Finite Fields
Decomposing a finite-dimensional commutative algebra over a finite field into its simple components, generalising polynomial factorisation.
Engineering / MathematicsMaximal Orders and Decomposition II2 min readKV-MATH-0616
Splitting a commutative algebra over a finite field into fields is the generalisation of polynomial factorisation, and it is the engine of the Buchmann-Lenstra decomposition method.
The setting
A finite-dimensional commutative algebra over a finite field, with no nilpotent elements, decomposes as a direct product of finite fields. The task is to find that decomposition explicitly.
The Berlekamp subalgebra generalises
Elements fixed by Frobenius form a subalgebra whose dimension equals the number of components. It is computed as a kernel.
Splitting a separable commutative algebra
- Remove nilpotentsQuotient by the radical if the algebra is not already reduced.
- Build the Frobenius matrixThe p-power map is linear over the prime field.
- Compute the kernelOf Frobenius minus the identity.
- Read the component countThe kernel dimension.
- SplitUse a non-trivial kernel element: its minimal polynomial factors, and the factors give idempotents separating the components.
Idempotents
The decomposition is equivalent to finding a complete set of orthogonal idempotents — one for each component. Each idempotent is obtained from a kernel element by evaluating a polynomial constructed from its minimal polynomial factors.
Large base fields
Application to prime decomposition
The order modulo p, quotiented by its radical, is exactly such an algebra. Its simple components correspond to the prime ideals above p, and their degrees are the residue degrees. Splitting the algebra therefore decomposes the prime.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 6.2.4. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.