Roberts, Mark (1981)
Normal forms, factorizations and eigenrings in free algebras.
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The rings considered in this thesis are the free algebras k〈X〉 (k a commutative field) and the more general rings Kk〈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 Kk〈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 Kk〈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),GLn (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.
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in Royal Holloway Research Online.Last modified on 01-Feb-2017
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Institution: University of London, Bedford College (United Kingdom).