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整体域.
整体域是代数数论研究的主要对象,分成两类:
- 数域:即有理数域 的有限扩张。
- 函数域:这里是指某个有限域 上有理函数域 的有限扩张。
整体域与局部域相对,整体域对一赋值作完备化便成为局部域。局部域上的分析较为简单;数学家通常先由局部域入手,再透过阿代尔环之构造研究整域情形。
戴德金与安里西·韦伯在19世纪末首先发现了数域与黎曼曲面的类比; 类比于复射影直线 ,有限扩张类比于分歧覆叠。安德烈·韦伊在1940年提出代数曲线的黎曼猜想,可视作此想法的进一步发展。
代数数论关心的课题原是数域,然而许多猜想或定理都有函数域上的类比,而后者技术上通常比较简单。因此,研究函数域有助于启示或厘清数域的情形。模型论上也有手法能将一些函数域的性质转移至数域。
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