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

设k为域，k为k的代数闭包。如果将a的基域k扩展到k，得到的k代数与k上的仿射线k同构，则可交换k代数a称为仿射线的k形。本课题的主要目的是研究仿射线的任意k形a的k-代数结构。在项目开始之前，只有一个零星的非平凡（即非多项式）仿射线的k-形式的例子是已知的。在该项目的两年期间，首席调查员获得了以下结果。首先，我们发现了一系列新的仿射线的非平凡k型，然后证明了仿射线的任意k型a与这些例子中的一个是k同构的。特别地，这样的k型a被给出为一个多项式环在k上的三变量余数模一个由三个元素生成的素理想P，它可以被显式地写出来（仿射线k型的结构定理）。作为这个定理的一个推论，我们有：仿射线的k型a是由k上的两个元素生成的，当且仅当上面定义的与a对应的素理想P是理想理论完全交。利用这些结果，我们还可以找到仿射线的k-形式的k-自同构群。为了证明这些结果，我们使用了伽罗瓦理论来证明环的扩展，这是首席研究员已经证明的。对这些结果的调查可以在[T.]浅沼，论A^1 -形式，富山大学教育教师回忆录第56期（2002年）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].