P 進數域
正文
又稱局部數域,它是數域關於p 進絕對值的完備化。p進數域的研究和代數數論的局部化方法,均始於K.亨澤爾1902年的工作。設p 是一個固定的素數,於是每個非零的有理整數α均可惟一地表成p 進位形式,即
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設K是任意代數數域,P為它的整數環OK中任一素理想。對每個非零元素
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局部數域Kp的有限次擴張仍是局部數域,於是有局部數域擴張的理論。局部數域有較簡單的代數結構和拓撲結構,而使得局部數域擴張理論較之於代數數域擴張理論要簡單。設l/K是局部數域的擴張,OK和OB分別是它們的整數環,P和B分別是OK和OB的惟一的極大理想,於是P在OB中生成的理想只能有形式Be,e稱為擴張l/K的分歧指數。另一方面,剩餘類域
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假設l/K是代數數域擴張,那么K中每個素理想P在l中有惟一的素因子分解式 PO,sub>L=
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1924年H.哈塞將這種思想成功地運用於二次型的研究之中。例如,設K 為代數數域,
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採用局部化方法(賦值論和Adèle、Idèle語言)能夠統一處理代數數域和以有限域為常數域的代數函式域。A.韋伊於1967年寫的《基礎數論》一書是這種方法的集中反映,對現代數論的發展有重要影響。