Research on the descent problem of base fields of open affine algebraic plane curves

设k是一个域，K是k的一个代数闭包。一个交换k-代数A称为仿射线的k-形式，如果A的基域k到K的扩张所得到的K-代数K-同构于K上的仿射线。本计画的主要目的是研究仿射直线的任意k-型A的k-代数结构。只有零星的例子非平凡（即非多项式）k-形式的仿射线已被称为开始之前的项目。在该项目的两年期间，首席调查员取得了以下成果。首先，我们找到了仿射线的一系列新的非平凡k-形式，然后证明了仿射线的任何k-形式A与其中一个例子k-同构。特别地，这样的k-形式A被给出为k上的三元多项式环模由三个元素生成的素理想P的剩余，其可以被显式地写出来（仿射线的k-形式的结构定理）。作为这个定理的推论，我们有：仿射线的k-形式A是由k上的两个元素生成的当且仅当对应于上面定义的A的素理想P是理想理论完全交。利用这些结果，我们也可以找到所有的群（直到同构）作为某个k-形式的仿射线的k-自同构群。为了证明这些结果，我们使用伽罗瓦理论的扩展环，这头调查员已被证明。这些结果的综述可以在[T.浅沼，关于A^1 -表格，富山大学教育学院回忆录第56期（2002）43-51]。

Let k be a field and let K be an algebraic closure of k. A commutative k-algebra A is called a k-form of an affine line if the K-algebra obtained by an extension of the base field k of A to K is K-isomorphic to an affine line over K. The main purpose of this project is to study k-algebraic structures of an arbitrary k-form A of an affine line. Only a sporadic examples of non trivial (i.e. non polynomial) k-forms of the affine line have been known before the start of the project. During a period of two years for the project the head investigator obtained the following results. First, we found a new series of non trivial k-forms of the affine line, and next proved any k-form A of the affine line is k-isomorphic to one of those examples. In particular, such a k-form A is given as a residue of a polynomial ring over k in three variables modulo a prime ideal P generated by three elements which can be explicitly written (Structure theorem of k-forms of affine line). As a corollary of this theorem, we have the following: A k-form A of the affine line is generated by two elements over k if and only if the prime ideal P corresponding A defined above is an ideal theoretic complete intersection. Using these results we can also find all groups (up to isomorphisms) obtained as the k-automorphism group of some k-form of an affine line. For the proof of these results, we use a Galois theory for extensions of rings, which the head investigator has been proved. A survey of these results can be found in [T. Asanuma, On A^1 -forms, Memoirs of the faculty of education Toyama University No.56 (2002)43-51].