F[x]-Modules: Similarity of t×t Matrices over a Field F

Abstract

The matrix A of a linear mapping α of a t-dimensional vector space relative to a basis, similarity classes, the characteristic polynomial χ _A (x). The F[x]-modules M(α) and M(A). Isomorphic modules M(A) are determined by similar matrices A and conversely. Order of a module element. Submodules of M(A). Direct sum of square matrices. General decomposition of M(A).

Cyclic R-modules, the F-basis $\mathcal{B}_{v_{0}}$ of the torsion F[x]-module 〈 v ₀ 〉, properties of companion matrices C(d(x)). Submodules and quotient modules of cyclic F[x]-modules. The Chinese remainder theorem for companion matrices, resultants.