Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.

Let D be a finite-dimensional division ring over its center Z(D), which is a finite field Fq. a) Prove that D is a finite field extension of Z(D) and determine its degree. Utilize Wedderburn's theorem in your analysis. b) Show that every finite division ring is commutative, thereby concluding that D is a field. Provide a detailed proof referencing relevant theorems. c) Investigate the structure of the multiplicative group D*. Determine its properties, such as being cyclic or abelian, and justify your conclusions. d) Explore the automorphism group Aut(D) of the division ring D. Describe its structure and determine whether all automorphisms are inner or if outer automorphisms exist. e) Examine the module theory over D by classifying all simple D-modules. Provide proofs for your classifications and discuss their significance in the broader context of module theory.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.4: Zeros Of A Polynomial

Problem 21E: Use Theorem to show that each of the following polynomials is irreducible over the field of...

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