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The Role of Divisible Groups in the Structure of Division Algebras and Brauer Group of Fields

Motiee Seyyed Mahalleh, Mehran | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 41224 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Hezavehi, Mohammad
  7. Abstract:
  8. Let D be an F-central division algebra of index n. Here, we present a criterion for the triviality of the group G(D) = D∗/NrdD/F (D∗)D′. In fact, we show that G(D) = 1 if and only if F∗2 = F∗2n and SK1(D) = 1 where SK1(D) is the reduced whitehead group of D. Then, we use this criterion to investigate the role of (non-abelian) divisible groups in the structure of division algebras. We prove that if D is not the ordinary quaternion algebra, D is similar to a tensor product of F-central cyclic algebras and D∗ contains a (non-abelian) divisible maximal subgroup, then D = F. In this direction, we observe that if the index of D is a prime, then D is a symbol algebra if and only if D∗ contains a non-abelian nilpotent subgroup. Furthermore, we study the structure of division algebras with radicable multiplicative
    groups. Recall that a division alegbra is called radicable if every equation of the form xn − a = 0 (a ∈ D∗ and n ∈ N) has a root in D. We prove that if F is indivisible, then the following statements are equivalent:
    • D is radicable;
    • D contains a divisible subfield K/F;
    • D is the ordinary quaternion algebra and F(√−1) is divisible.
    Moreover, if one of the above conditions holds, then F∗ is isomorphic to the multiplicative group of a real-closed field. Finally, we consider the structure of the Brauer group of an iterated n-fold Laurent series field. More precisely, let F be a field and Fn = F((x1)) . . . ((xn))
    be the field of iterated n-fold Laurent series with coefficients in F. By a theorem of Chang, if char(F) = 0 and F is an algebraically closed field, then Br(Fn)∼=⊕n(n−1)/2i=1 Q/Z. It is demonstrated that this result can be deduced from a group theoretic property of F∗. In the other way, we conclude that if char(F) = 0 and F∗ is divisible, then Br(Fn)∼= ⊕n(n−1)/2 i=1 Q/Z
  9. Keywords:
  10. Reduced Norm ; DIVISION GROUP ; Maximal Subgroup ; BRAUER GROUP ; Central Division Algebra

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