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In this thesis we review several arithmetical questions about the diophantine equation φ[x]=q where φ is a nondegenerate symmetric bilinear form on a finite dimensional vector space, φ[x] is the corresponding quadratic form and q is a nonzero element of the ground field. We obtain class number and mass formulae for the orthogonal group associated to φ in a number field utilizing the concept of quadratic forms, lattices and adelic language  

In this paper Alexander Berkovich and Hamza Yesilyurt revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2 + 5y2. Making use of Ramanujan’s 1 1 summation formula, they establish a new Lambert series identity for Σ1 n;m=1 qn2+5m2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But they do not stop there. Employing various formulas found in Ramanujan’s notebooks and using a bit of ingenuity, they obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x2+6y2, 2x2+3y2, x2+15y2, 3x2+5y2, x2+27y2, x2+5(y2+z2+w2), 5x2+y2+z2+w2. In the process, they... 

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... 

Gauss' theory of the arithmetic of quadratic forms appeared in Disquisitiones Arithmeticae (1801) and in particular Gauss presented a composition law for binary quadratic forms and related it to the arithmetic of quadratic extensions of Q. In the series of papers of “Higher Composition Laws” Manjul Bhargava gave a far reaching generalization of Gauss' composition law and extended it to binary cubic, and a number of other cases. He obtained six composition laws one of which the classical one of Gauss. Since the work of Gauss has had deep impact on number theory and in particular on the arithmetic of quadratic fields, one expects that Bhargava’s theory to lead to new insights in... 

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  

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... 

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...