The rings considered in this thesis are the free algebras k⟨X⟩ (k a commutative field) and the more general rings K<sub>k</sub>⟨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<sub>k</sub>⟨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<sub>k</sub>⟨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 H<sup>1</sup>(Gal (E/k),GL<sub>n</sub> (R⊗<sub> k</sub>E)) 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:704475 |
| Date | January 1981 |
| Creators | Roberts, Mark |
| Publisher | Royal Holloway, University of London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://repository.royalholloway.ac.uk/items/e865775d-e201-45ff-8ff5-a09fd054d096/1/ |
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