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P Fister’s Local-Global Principle

Nematollahi, Mohammad Ali | 2014

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 46588 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Gholamzadeh Mahmoudi, Mohammad
  7. Abstract:
  8. This master’s thesis has two major parts. The first part which includes chapters 1 to 8, describes the article A. Pfister, Quadratische Formen in beliebigen Körpern, Invent. Math. 1, (1966)pp. 116-132, and expresses important facts about the Witt ring W(K) of quadratic forms over an arbitrary field K of characteristic unequal to 2. Among those, it is shown that the order of each element in the additive group of W(K) is a power of 2, the Witt ring doesn’t have any zero divisor of odd dimension and a necessary and sufficient condition for W(K) to be an integral domain is given. The connections between the square class number of a field and the cardinality of its Witt ring and, providing some inequalities which hold between various invariants are other parts of this note. Besides, we study the class of even-dimensional quadratic forms which form a aximal ideal in W(K) and is denoted by M and, it is shown that M2 coincides with the subset of all elements of M whose discriminants are trivial.Also, M3 coincides with the subset of all elements of M2 whose class of associated Clifford algebra in the Brauer group of K is trivial.Beside the above it is shown that the level of every non formally real field is a power of 2 and for each n 2 N, an example of a field of level 2n is given. The most fundamental theorem of this part is the so-called Pfister’s local-global principle which provides a complete classification of torsion elements of the additive group of W(K).In second part, we express an application of Pfister’s local-global principle by which the relationship between Hilbert’s seventeenth problem and Pfister’s local-global principle is studied. Finally, with the aid of this principle, we express a proof of Hilbert’s seventeenth problem in the case of one variable. The remarkable feature of the proof is that we don’t use the fundamental theorem of algebra
  9. Keywords:
  10. BRAUER GROUP ; Clifford Algebra ; Quadratic Forms ; Pfister Forms ; Witt Ring ; Field Level ; Formally Real Field ; Pfister's Local-Global Principle ; Hilbert Seventeenth Problem

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  • کلیات
  • قضایای نمایش کسلز-فیستر
  • فرم‌های ضربی، سطح میدان
  • ویژگی‌های ساختاری حلقه‌ی ویت
  • ایده‌آل بنیادی حلقه‌ی ویت و رده‌ی براوئر جبر کلیفورد
  • اصل موضعی-سرتاسری فیستر
  • عدد رده‌ی مربعی، ناوردای یو، سطح میدان و عدد فیثاغورسی
  • رادیکال جیکوبسون و یکال‌های حلقه‌ی ویت
  • مسأله‌ی هفدهم هیلبرت
    • بحثی در رابطه با ترسیمات هندسی
    • حالتی خاص از مسأله‌ی هفدهم و اثباتی زیبا از هیلبرت
    • صورت کلی مسأله‌ی هفدهم هیلبرت
    • مسأله‌ی هفدهم هیلبرت و اصل موضعی-سرتاسری فیستر
  • کتاب نامه
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