The Arithmetic of Quadratic Forms
The Arithmetic of Quadratic Forms
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DOI：
10.1007/978-0-387-89486-7_7
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发表时间：
2009
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影响因子：
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通讯作者：
W. A. Coppel
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文献类型：
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作者：
W. A. Coppel
我们已经确定了可以表示为两个平方和的整数。类似地，人们可能会问哪些整数可以用x2 + 2 y2的形式表示，或者更一般地，用x2 + 2bxy+ cy 2的形式表示，其中a，B是给定的整数。算术理论的二元二次形式，其起源于工作的费尔马，是广泛发展的世纪期间，欧拉，拉格朗日，勒让德和高斯。扩展到二次型在两个以上的变量，这是开始由他们和例证拉格朗日定理，每一个正整数是一个总和的四个平方，是继续在19世纪世纪的狄利克雷，厄米特，H.史密斯，闵可夫斯基和其他人。在20世纪世纪哈塞和西格尔作出了显着的贡献。随着哈塞的工作，特别是它变得明显，理论是更加明确的，如果一个允许的变量是有理数，而不是整数。这为研究任意域上的二次型开辟了道路，维特（1937）和菲斯特（1965-67）做出了开创性的贡献。
We have already determined the integers which can be represented as a sum of two squares. Similarly, one may ask which integers can be represented in the formx2+ 2y2or, more generally, in the formax2+ 2bxy+cy2, wherea, b, care 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, Lagrange, Legendre and Gauss. The extension to quadratic forms in more than two variables, which was begun by them and is exemplified by Lagrange’s theorem 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. In the 20th century Hasse and Siegel made notable contributions. With Hasse’s work especially it became apparent that the theory is more perspicuous if 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).