Ebook Number theory - An introduction to mathematics (2/E): Part 2
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Ebook Number theory - An introduction to mathematics (2/E): Part 2
VIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask w Ebook Number theory - An introduction to mathematics (2/E): Part 2which integers can be represented in the form A2 + 2y2 or. more generally, in the form ax- + Ibxy + cv2. where a, b, (• are given integers. The arithmetic theory of binary quadratic forms, which had its origins in the work of Fermat, was extensively developed during the 18th century by Euler, Lagran Ebook Number theory - An introduction to mathematics (2/E): Part 2ge. Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange's theoremEbook Number theory - An introduction to mathematics (2/E): Part 2
that every positive integer is a sum of four squares, was continued during the 19th century by Dirichlet. Hermite. H.J.S. Smith. Minkowski and others.VIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask w Ebook Number theory - An introduction to mathematics (2/E): Part 2if one allows the variables to be rational numbers, rather than integers. This opened the way to the study of quadratic forms over arbitrary fields, with pioneering contributions by Witt (1937) and Pfister (1965-67).From this vast theory we focus attention on one central result, the Hasse-Minkowski Ebook Number theory - An introduction to mathematics (2/E): Part 2theorem. However, we first study quadratic forms over an arbitrary field in the geometric formulation of Witt. Then, following an interesting approachEbook Number theory - An introduction to mathematics (2/E): Part 2
due to Frohlich (1967), we study quadratic forms over a Hilbert field.1 Quadratic SpacesThe theory of quadratic spaces is simply another name for theVIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask w Ebook Number theory - An introduction to mathematics (2/E): Part 2en at quite an elementary level. The new approach had its debut in a paper by Witt (1937) on the arithmetic theory of quadratic forms, but it is appropriate also if one is interested in quadratic forms over the real field or any other field.For the remainder of this chapter ivc will restriel attenti Ebook Number theory - An introduction to mathematics (2/E): Part 2on to fields for which 1 + 1 / 0. Thus the phrase ’an arbitrary field' will mean ‘an arbitrary field of characteristic 2'. The proofs of many resultsEbook Number theory - An introduction to mathematics (2/E): Part 2
make essential use of this restriction on theW.A. Coppel. Number Theory. Ail Introduction to Mathematics^ Universitext.291292VII The Arithmetic of QuaVIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask w Ebook Number theory - An introduction to mathematics (2/E): Part 2roup Fx2 and any coset of this subgroup is called a square class.Let V be a finite-dimensional vector space over such a field F. We say that V is a quadratic space if with each ordered pair u. D of elements of V there is associated an element (w, t>) of F such that(i)(W| + 112' v) = (u\.v) + (112. t Ebook Number theory - An introduction to mathematics (2/E): Part 2>) for all W|, 112. V G V;(ii)(au. v) = a(u, d) for every a € F and all u, V G V;(iii)(u, u) = (u, m) for all w, I) G V.It follows that(i)' (w, t>| +Ebook Number theory - An introduction to mathematics (2/E): Part 2
1)2) = (m, U|) 4- (m, I>2) f°r all u, D|, 1)2 G V;(ii)' (u. at)) = a(u, D) for every a G F and all u, t) G V.Let ,..., en be a basis for the vector spVIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask w Ebook Number theory - An introduction to mathematics (2/E): Part 2'k=lwhere a;* = (ej.ek) = MA/. Thusn(u.u) = 22 ajkỉjtk j.k=iis a quadratic form with coefficients in F. The quadratic space is completely determined by the quadratic form, since(w, D) = {(u + I), u + d) - (u. u) - (t), w)J/2.-1Conversely, for a given basis e\....,e„ of V. any II X II symmetric matri Ebook Number theory - An introduction to mathematics (2/E): Part 2x A = (MỹA-) with elements from F. or the associated quadratic form f(x) = x'Ax. may be used in this way to give V the structure of a quadratic space.Ebook Number theory - An introduction to mathematics (2/E): Part 2
Let e\,e'„ be any other basis for V. Thenei = X TJier7=1where T = (rợ) is an invertible n X n matrix with elements from F. Conversely, any such matrixVIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask wVIIrhe Arithmetic of Quadratic FormsWe have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask wGọi ngay
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