埃米尔·阿廷Emil Artin

埃米尔·阿廷的父亲，也叫阿廷，是一位艺术品经销商。阿廷的母亲是Emma Laura-阿廷，她是一位歌剧演员。阿廷一生都热爱音乐，这种热爱基本上与他对数学的热爱相当。他在波希米亚的Reichenberg镇长大，当时该镇是奥地利帝国的一部分。虽然这个镇今天叫Liberec，位于捷克共和国北部，但在阿廷在那里接受教育的时候，它是一个主要讲德语的城市。作为纺织工业的中心，它常被昵称为“波希米亚的曼彻斯特”。

阿廷的童年并不是特别快乐，他后来在生命中回忆说自己曾感到孤独。与大多数数学家似乎发生的情况相反，他在很小的时候并没有发现自己被数学吸引，直到十六岁，这门学科对他来说并不比其他学校科目更重要，甚至不如某些科目。相当令人惊讶的是，直到十六岁，他都没有在这门学科上表现出任何特别的才能；至少这是他自己后来在生命中谈到学生时代时的看法。他确实表现出才能并且最被吸引的学校科目是化学。他在法国度过了一个快乐的学年，这是他学生时代最快乐的一年，然后在学校的最后两年里，他的兴趣转向了数学。

到1916年在Reichenberg参加学校毕业考试时，欧洲已经经历了两年第一次世界大战。然而阿廷确实开始了他的大学生涯，入读维也纳大学。然而，一个学期后，他被征召入奥地利军队，并在该军队服役直到战争结束。然后，在1919年1月，他进入莱比锡大学，在那里跟随古斯塔夫·赫格洛茨继续他的数学研究。学术成功很快到来，1921年他获得了博士学位。他的学位论文涉及将二次数域理论的方法应用于一个变量的有理函数在有限素数常数域上的二次扩张。获得博士学位后，他在哥廷根大学学习了一年（1921-22）。他于1922年10月前往汉堡大学担任助理，以开始1922-23学年冬季学期。1923年他获得了教授资格论文（Habilitation），并相应地成为汉堡的Privatdozent。

阿廷讲授了广泛的主题，包括数学、力学和相对论。他于1925年被提升为那里的编外教授，次年成为正式教授。这些年对阿廷的研究来说特别多产。理查德·布饶尔在[8]中写道：-

阿廷生命中1921-1931年的十年间[见证了]一种在数学家的生活中不常被匹敌的活跃。

他对域论、辫子理论做出了重大贡献，并且在1928年前后，他研究了关于右理想极小条件的环，现在称为阿廷环。他有一个殊荣，即在1927年解决了大卫·希尔伯特在1900年提出的23个著名问题之一。同样在1927年，他给出了一个一般互反律，它包含了自卡尔·弗里德里希·高斯产生他的第一条互反律以来所发现的所有已知互反律。

域论由施泰尼茨在1910年创立。它在接下来的十年中迅速发展，当阿廷在1924年解决以下问题时，他正遵循该主题的自然进展。他解决的问题是：给定一个代数闭域$O$，是否存在真包含于$O$中的子域$K$，使得$O$是其子域$K$的有限次代数扩张。在他1924年对这一问题的攻克中，阿廷仅限制自己考虑有理的域的代数闭包。然而，两年后的1926年，他意识到他的论证实际上证明的比他最初想的更多，并且他能够解决任何特征为0的代数闭域的问题。到这一阶段，他已经通过非常巧妙的论证，利用伽罗瓦理论和奥古斯丁·路易·柯西关于素数阶子群的定理，证明了$O$必须是$K$的2次扩张，并且子域$K$必须具有-1不能表示为平方和的性质。1926年，阿廷发表了一篇关于与Otto Schreier合作的重要论文，我们将在下面给出一些细节。

在进一步考察阿廷与Schreier1926年的合作论文之前，我们注意到这两人在1927年发表了一篇论文，其中他们能够处理上述问题在素特征域的情形。在这项1927年的工作中，他们引入了今天所谓的阿廷-Schreier次数为$p$的循环扩张。事实上，在素特征的情形下，他们证明了域$O$不可能是真子域$K$的有限扩张。

阿廷和Schreier早期的研究引导他们定义了今天所谓的形式实域，即具有-1不能表示为平方和性质的域。他们还定义了实闭域为那些形式实域，但其任何代数扩张都不是形式实域。阿廷本人证明了当$O$是代数数域时，实代数数的子域$K$解决了问题，而且，在域$O$的自同构意义下，它是唯一解。阿廷和Schreier在他们著名的1926年论文中发表了对所有形式实域和实闭域的研究，表明可以在它们上定义一种特定的序。既然已经建立了与有序域的联系，阿廷能够应用这些方法来解决大卫·希尔伯特的第17问题。阿廷在论文Über die Zerlegung definiter Funcktionen in Quadrate Ⓣ（《论定函数分解为平方和》）中给出了完整的解答，该论文也发表于1927年。还值得注意的是，实闭域理论直接影响了亚伯拉罕·鲁滨逊在模型论方面的贡献，特别是对于模型完全性和模型完备性的概念，例如参见[5]。

引导阿廷走向他的互反律的道路始于他仍是学生时。1920年，高木贞治发表了他关于类域论的基础论文，其中他围绕一个他发现的显著事实构建了理论，即由海因里希·马丁·韦伯定义的、在固定基域$k$上的类域集合与$k$上的阿贝尔扩张域集合相同。阿廷推进了高木贞治的工作，迈出了几个重大步骤。他定义了一种新型的$L$-级数，它推广了约翰·彼得·古斯塔夫·勒热纳·狄利克雷的$L$-级数，但性质相当不同。1923年，在Über eine neue Art von L-Reihen Ⓣ（《论一种新型L-级数》）中，阿廷能够获得他心中显然正在形成的那些结果的特例，这些特例依赖于现有互反律的使用。然而，1927年，他发表了关于该主题的杰作Beweis des allgemeinen Reziprozitätsgesetzes Ⓣ（《一般互反律的证明》），此时他以不同的方式发展了这些结果。

这个新思想起源于Nikolai 尼古拉·切博塔廖夫在1924年发表的工作，他在其中证明了一个由费迪南德·格奥尔格·弗罗贝尼乌斯提出的关于正规扩域的素理想集的密度的猜想。对阿廷的理论来说，被视为如此重要的并不是尼古拉·切博塔廖夫的结果，而是他在证明中使用的一种方法。以这个思想为基础，阿廷能够逆转他1923年的方法。阿廷没有使用现有的互反律，而是基于新方法证明了他的定理，这随后产生了一个包含所有先前互反律的新互反律。阿廷1927年论文中的定理已成为阿贝尔类域论的核心结果。Roquette写道[12]：-

在我看来，阿廷的互反律的主要重要性在于它开启了看待那些经典定律的新视角，将其表述为一个同构定理。情况类似于埃瓦里斯特·伽罗瓦理论，今天它是在抽象代数的框架中表述的，并以这种形式开启了新的应用和推广。类似地，阿廷的互反律为新的应用和进展开辟了道路。最引人注目的应用是由菲利浦·富特文勒给出的类域论主理想定理的证明，这是在阿廷的互反律发表一年后给出的。

阿廷在汉堡的第一个时期所做的另一项重要工作是辫子理论，他在1925年提出了这一理论。他再次通过引入这个新的研究领域展示了他的独创性，如今这个领域正被越来越多在group theory、semigroup theory和拓扑学中工作的数学家所研究。

阿廷提出了若干在数学发展中起了重大作用的猜想。其中两个由Roquette在[12]中提到，具有广泛的兴趣，即：-

第一，有限域上曲线的ζ函数的波恩哈德·黎曼猜想的类似物。在其开创性的博士学位论文中，阿廷在若干情形下通过数值验证了这一点。1933年，哈塞成功地对椭圆曲线证明了这一点，1942年，安德烈·韦伊对任意曲线证明了这一点。后来，众所周知，皮埃尔·德利涅将其推广到任意簇。因此，阿廷的这个猜想是现在所谓的算术几何中广泛活动的起源。

第二，有阿廷关于原根的猜想。给定任何不等于1或-1的整数g，且g不是某个其他整数的幂，那么阿廷猜想存在无穷多个素数p，使得g在卡尔·弗里德里希·高斯的意义下是模p的原根。更确切地说：这些素数的集合具有正密度，可以写下来并明确计算。阿廷于1927年9月27日向哈塞提出了这个猜想（根据哈塞日记中的一条记录），从那时起许多数学家试图证明它。Hooley在波恩哈德·黎曼假设（对数域）的一种强形式成立的条件下证明了它。托马斯·利特尔·希思-Brown和其他人证明了非常有趣的无条件结果。再次，阿廷的猜想在数论中引发了许多有趣的活动。

阿廷于1929年与他的一个学生Natalie Jasny结婚[8]：-

他的家庭现在在他的生活中占据了中心位置。当他的孩子们成长时，他在他们教育的各个阶段都积极参与。他每天花几个小时与他们在一起，对他来说，最重要的是向他们灌输他自己的个人和文化标准。

1933年1月30日希特勒上台，1933年4月7日的《公务员法》提供了将犹太教师从大学中清除的手段，当然也提供了将犹太裔人士从其他职位上清除的手段。所有非雅利安血统的公务员（只要有一位祖父母信奉犹太教，此人即被视为非雅利安人）都应被退休。阿廷不是犹太人，不受这些法律的影响。然而他的妻子是犹太人，因此当1937年《新官员法》通过时，那些与犹太人有婚姻关系的人受到了影响。阿廷[8]：-

……以他对个人自由的情感、他的正义感、他对肉体暴力的憎恶……

除了离开德国别无真正选择。1937年他移居美国，并在那里的多所大学任教。1937-38学年他在圣母大学，1938年至1946年在布卢明顿的印第安纳大学度过了八年，然后1946年至1958年在普林斯顿度过了十二年。

在美国期间，阿廷将精力投入教学和指导他的博士生，这些学生后来自己也做出了重大贡献。他发表的论文相对较少，但他写了一些极其重要的文本，已成为经典。1944年，他对右理想满足极小条件的环（现称为阿廷环）做了基础性工作。他对有理数上的半单代数提出了新的见解。1955年，他发表了两篇关于有限单群的重要论文，证明了已知（1955年）有限单群的阶中唯一重合的情形是由伦纳德·尤金·迪克森在其Linear groups中给出的。这项重要工作是导致对有限单群产生强烈兴趣的若干结果之一，最终导致了它们的分类。

阿廷的主要著作有Galois theory（1942年）、与C J Nesbitt和R M Thrall合著的Rings with minimum condition（1948年）、Geometric algebra（1957年）以及与约翰·泰特合著的Class field theory（1961年）。

也许他关于教学和写作教材的观点最好用他1953年写的一篇评论中的引文来说明：-

我们都相信数学是一门艺术。一本书的作者、课堂上的讲师试图向他的读者、他的听众传达数学的结构之美。在这种尝试中，他必定总是失败。数学当然是逻辑的，每个结论都是从先前推导出的陈述中得出的。然而它的整体，真正的艺术品，不是线性的；更糟的是，对它的领悟应当是瞬间的。我们都曾在某个罕见的场合体验过那种兴奋感，意识到我们使听众在一瞬间看到了整个架构及其所有分支。

1958年阿廷回到德国，再次被任命到汉堡大学，二十多年前他曾在如此不幸的情况下离开那里。他在1956年决定返回德国，因为在那一年他第一次休了学术休假，并在德国度过。这是他自1937年在纳粹控制下离开该国以来第一次访问该国。在学术休假期间，他重访了那些在他的数学成就中具有特殊地位的大学。他在哥廷根教了一个学期，然后回到汉堡，也在那里教了一个学期。1958年阿廷回到汉堡，在[8]中一段感人的文字里，理查德·布饶尔描述了1958年11月与阿廷一起走过汉堡街道的情景：-

一天下午，我们长时间散步，谈论着过去的时光。那是所有北方港口城市在深秋时节都熟知的雾蒙蒙、忧郁而相当阴郁的日子之一。我们无休止地在街道上徘徊，寻找着什么，我自己也不知道，直到我意识到，那是一个已不复存在的汉堡和永远逝去的时代。我相信，在阿廷的眼前，一定浮现出三十年前走过同样街道的年轻阿廷的形象，充满生机与力量。

然而，阿廷在数学之外有许多兴趣，他热爱化学、天文学和生物学。他也热爱音乐，是一位有造诣的音乐家，会演奏长笛、大键琴和古钢琴。Roquette写道[12]：-

我记得在汉堡时，他有一次告诉我他参加过一个电子音乐会议。

他是一位业余天文学家，甚至作为爱好自己建造了望远镜。

在[1]中，他的影响被描述如下：-

阿廷的科学成就只部分体现在他的论文和教科书以及他的讲座草稿中，这些草稿常常包含新的见解。它们也体现在他对同时代许多数学家的影响上，尤其是他的博士生（汉堡十一名，布卢明顿两名，普林斯顿十八名）。

Emil Artin's father, also called Emil Artin, was an art dealer. Emil's mother was Emma Laura-Artin and she was an opera singer. All his life Emil would have a love of music which essentially equalled his love of mathematics. He was brought up in the town of Reichenberg in Bohemia which was then part of the Austrian Empire. Although the town today is called Liberec, and is in the northern Czech Republic, at the time that Emil was educated there it was a mainly German speaking city. As the centre of the textile industry it was often nicknamed the "Bohemian Manchester".

Artin's childhood was not a particularly happy one and he recounted later in his life how he had felt lonely. He did not find himself attracted to mathematics at a very young age, contrary to what seems to happen to most mathematicians, and up to the age of sixteen the subject meant no more to him than any of his other school subjects and less than some. Rather surprisingly up to sixteen he did not show any particular talent for the subject; at least this was his own view of his schooldays when he spoke of them later in his life. The school subject which he did show talent for, and which he was most attracted towards, was chemistry. He spent a happy school year in France, the happiest of his schooldays, then his interests moved towards mathematics during his final two years at school.

By the time he took his school leaving examinations in 1916 in Reichenberg, Europe had already suffered two years of World War I. However Artin did begin his university career, enrolling at the University of Vienna. After one semester, however, he was drafted into the Austrian army and he served with this army until the end of the War. Then, in January 1919, he entered the University of Leipzig where he continued his mathematics studies with Herglotz. Academic success came quickly and in 1921 he was awarded his doctorate. His thesis concerned applying the methods of the theory of quadratic number fields to quadratic extensions of a field of rational functions of one variable taken over a finite prime field of constants. After receiving his doctorate he attended the University of Göttingen for one year (1921-22). He went to the University of Hamburg as an assistant in October 1922 for the start of winter semester of session 1922-23. In 1923 he had his Habilitation and accordingly became Privatdozent at Hamburg.

At Hamburg Artin lectured on a wide variety of topics including mathematics, mechanics and relativity. He was promoted to extraordinary professor there in 1925, then he became an ordinary professor in the following year. These were particularly productive years for Artin's research. Brauer wrote in [8]:-

The ten year period 1921-1931 of Artin's life [saw] an activity not often equalled in the life of a mathematician.

He made a major contribution to field theory, the theory of braids and, around 1928, he worked on rings with the minimum condition on right ideals, now called Artinian rings. He had the distinction of solving, in 1927, one of the 23 famous problems posed by Hilbert in 1900. Also in 1927 he gave a general law of reciprocity which included all previously known laws of reciprocity which had been discovered from the time that Gauss produced his first law.

Field theory had been created by Steinitz in 1910. It developed rapidly in the following decade and when Artin solved the following problem in 1924 he was following the natural progression for the topic. The problem which he solved was whether, given an algebraically closed field $O$, there exist subfields $K$, properly contained in $O$, with $O$ an algebraic extension of finite degree of its subfield $K$. In his 1924 attack on this problem Artin restricted himself to considering only fields which were an algebraic closure of the field of rationals. However, two years later in 1926 he realised that his arguments actually proved more than he had originally thought, and he was able to solve the problem for any algebraically closed field of characteristic 0. By this stage he had proved, using very clever arguments with Galois theory and Cauchy's theorem on subgroups of prime order, that $O$ had to be an extension of $K$ of degree 2 and that the subfield $K$ had to have the property that -1 could not be expressed as a sum of squares. In 1926 Artin published an important paper on joint work with Otto Schreier and we give some details below.

Before looking further at the joint 1926 paper of Artin and Schreier we note that the pair published a 1927 paper in which they were able to handle the problem described above in the case of fields of prime characteristic. In this 1927 work they introduced what are called today Artin-Schreier cyclic extensions of degree $p$. In fact, in the case of prime characteristic, they proved that the field $O$ cannot be a finite extension of a proper subfield $K$.

The earlier research by Artin and Schreier had led them to define what today are called formally real fields, they are fields with the property that -1 cannot be expressed as a sum of squares. They also defined real-closed fields to be those that were formally real yet every algebraic extension of them failed to be formally real. Artin himself proved that when $O$ is the field of algebraic numbers, the subfield $K$ of real algebraic numbers solves the problem and, moreover, it is the unique solution up to automorphisms of the field $O$. Artin and Schreier published in their famous 1926 paper their studies of all formally real fields and real closed fields, showing that a specific ordering could be defined on them. Now that the connection had been made with ordered fields, Artin was able to apply these methods to solve Hilbert's 17th problem. Artin gave a complete solution in the paper Über die Zerlegung definiter Funcktionen in Quadrate Ⓣ also published in 1927. It is also worth noting that the theory of real-closed fields directly influenced Abraham Robinson in his contributions to model theory, particularly for the concepts of model completeness and model completion, see for example [5].

The path which led Artin to his reciprocity law began while he was still a student. In 1920 Takagi published his fundamental paper on class field theory in which he built the theory around a remarkable fact which he had discovered, namely that the set of class fields, as defined by Heinrich Weber, over a fixed ground field $k$ is identical to the set of abelian extension fields over $k$. Artin took the work of Takagi forward making several major steps. He defined a new type of $L$-series, which generalised Dirichlet's $L$-series, yet was quite different in nature. In 1923 in Über eine neue Art von L-Reihen Ⓣ Artin was able to obtain special cases of the results which were clearly forming in his mind and these special cases depended on the use of existing reciprocity laws. However, in 1927 he published his masterpiece on the subject Beweis des allgemeinen Reziprozitätsgesetzes Ⓣ where now he had developed the results differently.

The new idea originated in work which Nikolai Chebotaryov published in 1924 where he had proved a conjecture made by Frobenius about the density of the set of prime ideals of a normal extension field. It was not Chebotaryov's result which was seen to be so important for Artin's theories, rather it was a method he used in his proof. With this idea as a basis Artin was able to reverse his 1923 approach. Instead of using the existing reciprocity laws, Artin proved his theorems based on the new approach which then yielded a new reciprocity law which contained all previous reciprocity laws. The theorems of Artin's 1927 paper have became central results in abelian class field theory. Roquette writes [12]:-

In my opinion, the main importance of Artin's Reciprocity Law is that it opens a new viewpoint on those classical laws, formulating it as an isomorphism theorem. The situation is similar to that with Galois Theory which, today, is formulated in the framework of abstract algebra, and in this form opens new applications and generalizations. Similarly, Artin's Reciprocity Law opens the way to new applications and progress.The most striking application was given by Furtwängler's proof of the principal ideal theorem of class field theory, given one year after the publication of Artin's Reciprocity Law.

Another important piece of work done by Artin during his first period in Hamburg was the theory of braids which he presented in 1925. He again showed his originality by introducing this new area of research which today is being studied by an increasing number of mathematicians working in group theory, semigroup theory, and topology.

Artin made a number of conjectures which have played a large role in the development of mathematics. Two of these, mentioned by Roquette in [12], have wide interest, namely:-

First, the analogue of the Riemann conjecture for the zeta function of a curve over finite fields. In his seminal Ph.D. thesis Artin verified this in a number of cases numerically. In 1933 Hasse succeeded in proving this for elliptic curves, and in 1942 Weil for arbitrary curves. Later, as is well known, Deligne generalized this for arbitrary varieties. Thus, this conjecture of Artin was the origin of a wide range of activities in what is now called arithmetic geometry.

Second, there is Artin's conjecture on primitive roots. Given any integer g not 1 or -1, and g not a power of some other integer, then Artin conjectured that there are infinitely many prime numbers p such that g is a primitive root modulo p in the sense of Gauss. More precisely: the set of those prime numbers has positive density, which can be written down and computed explicitly. Artin made this conjecture to Hasse on 27 September 1927 (according to an entry in Hasse's diary), and since then many mathematicians have tried to prove it. Hooley has proved it under the condition that a strong form of Riemann's hypothesis (for number fields) is valid. There are very interesting unconditional results, proved by Heath-Brown and others. Again, Artin's conjecture triggered a lot of interesting activities in number theory.

Artin married one of his students, Natalie Jasny, in 1929 [8]:-

His family now occupied a central position in his life. When his children were growing up, he took a most active part in all phases of their education. He spent hours with them every day, and it was of foremost importance to him to instil in them his own personal and cultural standards.

On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Artin was not a Jew and was not affected by these laws. However his wife was a Jew so when the 'New Official's Law' was passed in 1937 those related to Jews by marriage were affected. Artin [8]:-

... with his feeling for individual freedom, his sense of justice, his abhorrence of physical violence ...

had no real alternative but to leave Germany. In 1937 he emigrated to the United States and taught at various universities there. He was at Notre Dame for the academic year 1937-38, he spent eight years at Bloomington at Indiana University from 1938 to 1946, and then he was twelve years at Princeton from 1946 to 1958.

During his years in the United States Artin put his energies into teaching and supervising his Ph.D. students who themselves went on to make major contributions. He published relatively few papers, but he wrote a number of extremely important texts which have become classics. In 1944 he did fundamental work on rings with the minimum condition on right ideals, now called Artinian rings. He presented new insight into semi-simple algebras over the rationals. In 1955 he produced two important papers on finite simple groups, proving that the only coincidences in orders of the known (in 1955) finite simple groups were those given by Dickson in his Linear groups. This important piece of work is one of a number of results leading to the intense interest in finite simple groups which eventually led to their classification.

Among Artin's main books are Galois theory (1942), Rings with minimum condition (1948) written jointly was C J Nesbitt and R M Thrall, Geometric algebra (1957) and Class field theory (1961) written with J T Tate.

Perhaps his views on teaching and writing texts are best illustrated by a quotation from a review he wrote in 1953:-

We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail. Mathematics is logical to be sure, each conclusion is drawn from previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that, its perception should be instantaneous. We have all experienced on some rare occasion the feeling of elation in realising that we have enabled our listeners to see at a moment's glance the whole architecture and all its ramifications.

In 1958 Artin returned to Germany, being appointed again to the University of Hamburg which he had left in such unhappy circumstances over 20 years before. He made the decision to return to Germany in 1956, for in that year he took his first ever sabbatical leave which he spent in Germany. It was his first visit to that country since he left it in the grip of the Nazis in 1937. During his sabbatical leave he revisited universities which had a special place in his mathematical achievements. He taught for a term in Göttingen and then returned to Hamburg where he also taught for a term. In 1958 Artin returned to Hamburg and, in a moving passage in [8], Brauer describes walking through the streets of Hamburg with Artin in November 1958:-

We took a long walk one afternoon talking of old times. It was one of those misty, melancholy, and rather miserable days which all northern harbour cities know so well in late fall. We wandered endlessly through the streets searching, I did not know for what, until I realised, it was a Hamburg which no longer existed and times which were gone for ever. Before Artin's eyes, I believe, there must have been the picture of the young Artin who had walked through the same streets thirty years before, full of life and strength.

Artin had many interests outside mathematics, however, having a love of chemistry, astronomy and biology. He also loved music and was an accomplished musician playing the flute, harpsichord and clavichord. Roquette writes [12]:-

I remember in Hamburg when he once told me of a conference on electronic music which he had attended.

An amateur astronomer, he even built his own telescope as a hobby.

In [1] his influence is described as follows:-

Artin's scientific achievements are only partially set forth in his papers and textbooks and in the drafts of his lectures, which often contain new insights. They are also to be seen in his influence on many mathematicians of his period, especially his Ph.D. candidates (eleven in Hamburg, two in Bloomington, eighteen in Princeton).