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In this thesis, which is based on an article of Claus Scheiderer (Reference [21]), the relation between positive semi-definite elements of a semilocal ring, called psd, and elements of the ring which can be written as a sum of square elements of the ring and are called sos will be evaluated. In Chapter One, the concepts of prime cones, real spectrum of a ring and real ideals will be defined and then the relation between psd and sos elements in the total rings of quotients will be assessed. The Krull Valuation of an element which is a sum of squares will also be determined. In Chapter Two, the concept of totally positive elements in a ring will be defined and their properties will be... 

For a field of characteristic not two, the classical u-invariant is defined as the maximal dimension of anisotropic quadratic forms over F. Initially Kaplansky conjectured that u(F), when finite, is always a 2-power. Later Merkurjev constructed a field F such that u(F) = 6. This dissertation examines in detail the article: R. Elman, T. Y. Lam, Quadratic forms and the u-invariant. I. Math. Z. 131, 283-304 (1973). in which the notion of ”generalized u-invariant” (motivated by Pfister’s Local-Global Principle) was defined as the maximal dimension of anisotropic torsion quadratic forms over F. This is indeed a right generalization of the definition of the classical u-invariant since it not only... 

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... 

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... 

The Sato-Tate conjecture is an important conjecture regarding the distribution of the Frobenius traces of a family of elliptic curves over finite fields obtained from the reductions of an elliptic curve without CM over a number field modulo the prime ideals of its ring of integers. The statement is that the sequence of normalized Frobenius traces should follow a semicircle distribution. It was discovered by Mikio Sato and reformulated by John Tate in terms of L-functions around 1960. A complete proof of the conjecture for elliptic curves over totally real fields was published in 2008 by R. Taylor et al. under some mild technical assumptions. In addition to the original Sato-Tate conjecture,... 

Let D be an F-central division algebra of index n. In this thesis a criterion for the triviality of the group G(D) = D^*/Nrd_(D/F) (D^*)D^' is presented and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK〗_1 (D) = 1 and (F^* )^2=(F^* )^2n. By using this, the role of some particular subgroups of D* in the algebraic structure of D is investigated. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup  

In this thesis, we investigate the involutions of a Clifford algebra induced by involutions of orthogonal group in characteristic two. Several properties of these involutions, such as the relations between their invariants and their decompositions are studied. Also it is shown that a tensor product of quaternion algebras with involution can be expressed as the Clifford algebra of a suitable quadratic form with an involution induced by an involution of orthogonal group. Finally, in connection with the Pfister factor conjecture formulated by D. B. Shapiro, split tensor products of quaternion algebras with involution over a field of characteristic two are investigated