Normal forms, factorizations and eigenrings in free algebras

Abstract

The rings considered in this thesis are the free algebras k⟨X⟩ (k a commutative field) and the more general rings K_k⟨X⟩ (K a skew field and k a subfield of the centre of K) given by the coproduct of K and k⟨X⟩ over k. The results fall into two distinct sections.

The first deals with normal forms; using a process of linearization we establish a normal form for full matrices over K_k⟨X⟩ under stable association. We also give a criterion for a square matrix A over a skew field K to be cyclic---that is, for xI---A to be stably associated to an element of K_k⟨X⟩ (here k = centre(K)).

The second section deals with factorizations and eigenrings in free algebras. Let k be a commutative field, E/k a finite algebraic extension and P a matrix atom over k⟨X⟩. We show that if E/k is Galois then the factorization of P over E⟨X⟩ is fully reducible; if E/k is purely inseparable then the factorization is rigid. In the course of proving this we prove a version of Hilbert's Theorem 90 for matrices over a ring R that is a fir and a k-algebra; namely that H1(Gal (E/k),GL_n (R⊗ _kE)) is trivial for any Galois extension E/k. We show that the normal closure F of the eigenring of an atom p of k⟨X⟩ provides a splitting field for p (in the sense that p factorizes into absolute atoms in F⟨X⟩). We also show that if k is any commutative field and D a finite dimensional skew field over k then there exists a matrix atom over k⟨X⟩ with eigenring isomorphic to D.