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Units in Witt Rings

Karimi Dehkordi, Mehdi | 2016

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48609 (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 three chapters. In the first and second chapters provided all the necessary preparations for the third chapter to describes the following article: Lewis, D. W, Units in Witt rings, Commun. Algebra 18, no. 10, 3295-3306 (1990).The first chapter includes an introduction of quadratic forms and Witt ring on fields with characteristic unequal 2, studing W b(F) in the category of commutative rings and introduction of formally real and nonreal fields. In this chapter there are important theorems such as Witt’s Decomposition and Cancellation Theorem, Cassels Representation, Springer and Pfister’s Local-Global Principle. The second chapter introduces the discretely valuated fields and ultimately pays local and global fields. In this chapter we will introduce Hasse- inkowski theorem that has fundamental application in chapter three. Finally in chapter three we will describe the main topic, Units in Witt ring, such as we will study units on formally real and nonreal fields and will show that the order of each unit is 2 on finite, p-adic and algebraic number fields. We will show when field is formally real, these units are exactly equal to class of forms with signature equal ±1 and in case nonreal are equal to class of forms with odd dimension, also any unit in W(F) has finite order and the order is a power of two. We will show that if x 2 I, the fundamental ideal in W(F), is torsion with 2kx = 0 then (1 + x)2k = 1, moreover in the binomial expansion of (1 + x)2k each term after the first will vanish. With an example show the converse of this is not valid in general, and we shall prove if x 2 I2 and is torsion then the converse is true
  9. Keywords:
  10. Quadratic Forms ; Witt Ring ; Pfister's Local-Global Principle ; P-Adic Field ; Formally Real Field ; Local Field ; Global Field ; Nonreal Field ; Hasse-Minkowski Theorem

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