彼得·卢德维格·梅德尔·西罗Peter Ludwig Mejdell Sylow

西罗的父母是Thomas Edvard von Westen 西罗（1792-1875）和Magdalena Cecilie Cathrine 西罗（1806-1898）。Thomas Edvard Sylow曾是骑兵上尉，后来成为政府大臣。从1848年到1854年，他担任大臣兼陆军部部长。他出生于挪威北特伦德拉格的斯纳萨，是西罗 Ludvik von Westen 西罗和Elisabeth Christine Niemeyer的儿子。Magdalena Cecilie Cathrine 西罗出生于挪威奥普兰郡兰德的塞普顿，是Johan Ernst 西罗和Fredrikke Caroline Sophie Kierulf的女儿。彼得·卢德维格·梅德尔·西罗是他父母十个孩子中的长子，有弟弟妹妹Fredrikke Caroline Sophie Sylow（生于1834年）、Elisabeth Arnolda von Westen 西罗（生于1835年）、Johan Ernst 西罗（生于1837年）、Carl Christian Weinwich 西罗（生于1838年）、Sverre Thomas Sylow（生于1839年）、Halfdan 西罗（生于1840-56年）、Johanne Hermine 西罗（生于1843年）、Stephanie Martine 西罗（生于1844年）和Martin Kierulf 西罗（生于1848-58年）。这十个孩子并非都活到成年：Sverre Thomas在婴儿期夭折，Halfdan十六岁去世，Martin Kierulf十岁去世。

西罗有着良好的教养，学会了如何独立工作，被教导要尽力而为和努力工作的重要性，然而在某些方面，他的教养在他的职业生涯中却被证明是一种不利条件。这是因为他被教导要谦逊，尽管这是出于最好的意图，但这仍然意味着他乐于在比本应拥有的更低微的职位上度过许多年。有趣的是，1902年，当西罗在纪念尼尔斯·阿贝尔诞辰一百周年的会议上致欢迎辞时，他说（见[2]）：-

……如此伟大的谦逊也许不适合这个世界，它也可能被视为一种弱点。

西罗就读于克里斯蒂安尼亚大教堂学校，于1850年毕业。随后，他在克里斯蒂安尼亚大学开始学习自然科学，并于1853年赢得了一场数学竞赛。接着，他于1856年参加了高中教师资格考试，以优异的成绩获得了数学和自然科学教师资格。由于没有大学职位空缺，他在Hartvig Nissen学校教了两年书。之后，他搬到了东福尔郡的腓特烈斯哈尔德镇（现称哈尔登），从1858年到1898年在中学任教。遗憾的是，尽管西罗本可以成为一名杰出的大学讲师，但他并没有成为一名特别好的中学教师。他对数学的高级领域感兴趣，对较低层次的教学缺乏热情。此外，他发现很难维持课堂纪律，因此他的职业生涯主要在学校而非大学，这在两个方面都是对他才能的糟糕利用——大学因为没有西罗作为讲师而更贫乏，而学校因为有他作为教师而更贫乏！

然而，西罗继续着他的数学研究（见[5]）：-

……起初按照尼尔斯·阿贝尔和卡尔·古斯塔夫·雅各布·雅可比的传统研究椭圆函数，受到纯粹数学教授Ole Jacob Broch的启发。发现尼尔斯·阿贝尔关于代数方程可用根式求解的论文更有趣后，西罗（由应用数学教授卡尔·安东·皮耶克尼斯引导）从这些论文转向了埃瓦里斯特·伽罗瓦。

Ole Jacob Broch曾在克里斯蒂安尼亚大学学习数学，之后前往巴黎、柏林和柯尼斯堡，在那里研究了多个数学主题，尤其是光学和统计学。回到挪威后，他与朋友Hartvig Nissen（1815-1874）一起创办了Hartvig Nissen学校，这是克里斯蒂安尼亚Uranienborg区的一所独立女子学校。Broch于1848年成为克里斯蒂安尼亚大学的应用数学讲师，但在1850年Bernt Holmboe去世后接替了他的纯粹数学教学。Broch保持着对Hartvig Nissen学校的兴趣，他是该校的联合创始人之一，并给予年轻教师西罗很大鼓励，让他继续从事高等数学研究。尽管起初西罗觉得阅读尼尔斯·阿贝尔的论文很困难，但很快他发现尼尔斯·阿贝尔对方程理论的理解远比其已发表论文所显示的深刻。西罗最初试图发表一些他在尼尔斯·阿贝尔论文中发现的未发表结果时遇到了问题。这是因为他将包含这些结果的论文寄给了奥古斯都·利奥波德·克雷勒的Journal，该论文在那里由利奥波德·克罗内克考虑发表。此时困难出现了，因为利奥波德·克罗内克已经发表过这些结果，是他自己发现的，并且不希望有一篇论文表明尼尔斯·阿贝尔远在他之前就证明了这些结果。因此，利奥波德·克罗内克拒绝了西罗的论文，但当然，由于西罗正确地描述了尼尔斯·阿贝尔未发表的结果，最终这些结果确实被归功于尼尔斯·阿贝尔而非利奥波德·克罗内克。

1861年，西罗获得奖学金出国旅行，访问了柏林和巴黎。在巴黎，他听了米歇尔·沙勒关于theory of conics的讲座、约瑟夫·刘维尔关于理性力学的讲座以及让-马里·迪阿梅尔关于极限理论的讲座。他在奖学金结束时写的报告中说，他还：-

……熟悉了较新的著作，尤其是方程理论方面的。

在柏林，他与利奥波德·克罗内克进行了有益的讨论，但无法参加卡尔·魏尔斯特拉斯的课程，因为当时卡尔·魏尔斯特拉斯生病了。Bent Birkeland在[5]中写道，由于柏林没有他感兴趣的课程：-

……他转而到图书馆工作，研究数论和方程理论。他还结识了卡尔·威廉·博尔夏特，即‘奥古斯都·利奥波德·克雷勒杂志’的编辑……有趣的是，无论他在巴黎还是柏林逗留期间，都没有提到任何代数或方程理论的讲座。

1862年，西罗在克里斯蒂安尼亚大学讲学，替代已当选为挪威议会Storting成员的Broch。在他的讲座中，西罗解释了尼尔斯·阿贝尔和埃瓦里斯特·伽罗瓦关于代数方程的工作。这些讲座的摘要见[4]。值得注意的是，尽管当时他尚未证明“西罗定理”（他10年后才发表），但他确实对它们提出了一个问题。在证明了奥古斯丁·路易·柯西定理——一个阶可被prime$p$整除的群有一个阶为$p$的子群——之后，西罗问它能否推广到$p$的幂。这些讲座之所以重要有几个原因，尤其是当时听讲的一名年轻学生是索菲斯·李。西罗的讲座极有价值，使Lie对一个他后来将作出重大贡献的课题有了根本性的理解。Broch于1865年至1868年再次在Storting任职，他热切希望西罗在此期间接替他的大学教学。然而，Fredrikshald的学校拒绝给西罗去大学任教的假期。1869年，Broch离开他的纯数学讲席，留下一个西罗完全有资格填补的空缺。然而，克里斯蒂安尼亚大学当时对纯数学评价不高，更偏好较实用和可应用的课题。西罗的方法过于理论化，因此他没有被任命。

1870-71年，西罗与朱利叶斯·佩特森交换了九封信，后者当时正在撰写他的学位论文。朱利叶斯·佩特森就他学位论文的主要定理征求西罗的意见，这些信全都涉及此事。文章[8]讨论了这些信。这两位数学家在1876-77年又交换了十六封信。1873年至1881年间，西罗和索菲斯·李准备了一版尼尔斯·阿贝尔全集，以Oeuvres complète de Niels Henrik AbelⓉ（阿贝尔全集）为题出版。此事的动因来自挪威科学院，它向挪威议会申请该项目经费，很快获得批准。这笔经费使西罗得以离开学校四年，专心投入该项目。Arild Stubhaug在[3]中写道：-

在工作过程中，西罗有一个热心的讨论者卡尔·安东·皮耶克尼斯，后者则致力于1880年出版的尼尔斯·阿贝尔传记。卡尔·安东·皮耶克尼斯希望尼尔斯·阿贝尔的早期著作尽可能多地出版，不仅是他那些具有典范严谨性的伟大论著，而且也许还有比Holmboe在其1839年版中所用的更多的尼尔斯·阿贝尔材料？

在1881年12月9日出版的西罗/索菲斯·李版中，发现并出版了许多额外的尼尔斯·阿贝尔材料。索菲斯·李说大部分工作是由西罗完成的，并评论说西罗因其以下方面而理应获得一个大学职位：-

……广博的知识、敏锐的批判力和杰出的数学工作。

然而，今天西罗的声名建立在一篇1872年发表的10页论文上。在这篇西罗发表于Mathematische Annalen第5卷（第584至594页）的Théorèmes sur les groupes de substitutions中，出现了三个西罗定理，尽管我们知道他在1870年9月之前已经证明了他著名的定理。奥古斯丁·路易·柯西已经证明，一个阶可被素数$p$整除的群有一个阶为$p$的元素。西罗推广了这一点，证明了也许是有限群论中最深刻的结果：

如果$p^{n}$是整除群$G$阶的素数$p$的最大幂，那么

1. $G$有阶为$p^{n}$的子群
2. $G$有$1 + kp$个这样的子群
3. 任意两个这样的子群共轭。

几乎所有关于有限群的工作都使用西罗的定理。西罗1872年的原始论文由Rod Gow在[7]中讨论，[6]的作者们和[15]的作者也讨论过。在[11]中，Winfried Scharlau描述了西罗如何通过研究埃瓦里斯特·伽罗瓦的工作，特别是埃瓦里斯特·伽罗瓦关于素数次数方程可解性的判别法，而得出他的发现。论文[11]解释了西罗如何在证明中使用来自伽罗瓦理论的方法。

西罗成为Acta Mathematica的编辑，于1883年当选为哥廷根科学院成员，并于1894年获得哥本哈根大学的名誉博士学位。

索菲斯·李在克里斯蒂安尼亚大学为西罗设立了一个特别讲席，西罗从1898年起在该大学任教。注意，他获得大学职位时已经65岁，但值得注意的是，他仍能担任此职位20年。G A Miller写道[10]：-

尽管西罗进入大学教职时年事已高，据说他在二十年间以显著的成功担任该职位。直到他生命的最后一年，他才似乎感到教授职责繁重，那时他常说自己感到疲倦。

虽然他在65岁时终于成为大学教授，但我们应该注意到，他并未获得教授的薪水。起初，他领取的是校长薪水，大约为大学教授薪水的一半。后来他的薪水有所增加。

1902年，西罗在纪念尼尔斯·阿贝尔诞辰一百周年的会议上致欢迎辞。他说（见[2]）：-

在十九世纪初，应用数学已经取得了巨大的成就，特别是在天文学和物理学领域。但就在同时，数学……开始将目光转回纯粹和抽象的理论。[卡尔·弗里德里希·高斯和奥古斯丁·路易·柯西]发起了那场伟大的运动，这场运动贯穿了整个上一个[19]世纪，它从基础上改革了数学，同时用新理论丰富了它。……正是在这场运动中，尼尔斯·阿贝尔发挥了如此重要的作用，以至于他将永远被列为有史以来最伟大的数学家之一。

1902年，西罗与Elling Holst合作，出版了尼尔斯·阿贝尔的通信集。在1881年西罗/索菲斯·李的书出版后，又发现了更多的尼尔斯·阿贝尔文件，并且在1913年于克里斯蒂安尼亚举行的“第三届斯堪的纳维亚数学家大会”上，西罗讨论了这些新材料。

西罗讨论的引言版本在THIS LINK

在[9]中，G A Miller引用了费迪南德·格奥尔格·弗罗贝尼乌斯对西罗的评价：-

[西罗]因其著名的以他命名的定理而为每个文明国家的数学家所知。1876年，费迪南德·格奥尔格·弗罗贝尼乌斯评论道：“正如每个受过教育的人都知道勾股定理一样，每个数学家都会谈到尼尔斯·阿贝尔定理和西罗定理”。

我们不应给人这样的印象：西罗定理和尼尔斯·阿贝尔材料是西罗仅有的数学贡献。他还发表了几篇关于椭圆函数的论文，特别是关于复乘法的论文，以及关于群论的论文。最后，我们应该稍微谈谈西罗数学之外的生活。他从未结婚，但是一个热情的人，具有良好的幽默感。他热爱户外活动，经常在山区度过暑假，通常在Kongsvoll，在那里他研究植物。Kongsvoll是奥斯陆和特隆赫姆之间路线上的一个山间驿站，提供食物和住所，建于该路线被前往特隆赫姆圣奥拉夫圣地的朝圣者使用时。

Ludwig Sylow's parents were Thomas Edvard von Westen Sylow (1792-1875) and Magdalena Cecilie Cathrine Mejdell (1806-1898). Thomas Edvard Sylow was a captain in the cavalry and later became a government minister. From 1848 to 1854 he was Minister and Chief of the Army Ministry. He was born in Snasa, Nord-Trondelag, Norway, the son of Peter Ludvik von Westen Sylow and Elisabeth Christine Niemeyer. Magdalena Cecilie Cathrine Mejdell was born in Septon, Land, Oppland, Norway, the daughter of Johan Ernst Mejdell and Fredrikke Caroline Sophie Kierulf. Peter Ludwig Mejdell Sylow was the eldest of his parents' ten children, having younger siblings Fredrikke Caroline Sophie Sylow (born 1834), Elisabeth Arnolda von Westen Sylow (born 1835), Johan Ernst Mejdell Sylow (born 1837), Carl Christian Weinwich Sylow (born 1838), Sverre Thomas Sylow (born 1839), Halfdan Sylow (born 1840-56), Johanne Hermine Sylow (born 1843), Stephanie Martine Sylow (born 1844), and Martin Kierulf Sylow (born 1848-58). Not all of these ten children reached adulthood: Sverre Thomas died as a baby, Halfdan died aged sixteen, and Martin Kierulf died aged ten.

Although Sylow had a good upbringing, learning how to work on his own, being taught the importance of doing one's best and working hard, nevertheless in some ways his upbringing would prove a disadvantage in his career. This was because he was taught to be modest and although this was done with the best of intentions, still it meant that he was happy to spend many years in a more lowly position than he should have had. It is interesting to note that in 1902, when Sylow gave the welcoming address at a conference to mark the centenary of Niels Abel's birth, he said (see [2]):-

... such a great modesty maybe does not fit this world, it may also be seen as a weakness.

Sylow attended Christiania Cathedral School, graduating in 1850. He then began his studies of natural sciences at Christiania University where won a mathematics contest in 1853. He then took the high school teacher's examination in 1856, qualifying as a teacher of mathematics and natural science with excellent grades. As no university post was available, he taught for two years at the at Hartvig Nissen school. He then moved to the town of Fredrikshald (now called Halden) in Ostfold county, where he taught at secondary school from 1858 to 1898. Sadly, although Sylow would have made an outstanding university lecturer, he did not make a particularly good school teacher. He was interested in the advanced areas of mathematics and had little enthusiasm for teaching at lower levels. Also he found it difficult to keep discipline in his classroom so the fact that his career was largely in schools rather than universities was a poor use of his talents on two scores - universities were the poorer for not having Sylow as a lecturer, while schools were poorer for having him as a teacher!

Sylow continued his mathematical studies however (see [5]):-

... at first working on elliptic functions in the tradition of Abel and Jacobi, inspired by the professor of pure mathematics Ole Jacob Broch. Finding Abel's papers on the solvability of algebraic equations by radicals more interesting, Sylow was led from them (by the professor in applied mathematics, Carl Bjerknes) to Galois.

Ole Jacob Broch had studied mathematics at Christiania University after which he travelled to Paris, Berlin and Königsberg where he studied a range of mathematical topics, particularly optics and statistics. After returning to Norway, together with his friend Hartvig Nissen (1815-1874), he founded the Hartvig Nissen school, an independent girls' school in the Uranienborg district of Christiania. Broch became a lecturer in applied mathematics at Christiania University in 1848, but took over Bernt Holmboe's pure mathematics teaching in 1850 following his death. Broch retained his interest in the Hartvig Nissen school, of which he had been a co-founder, and gave the young teacher Sylow much encouragement to continue his advanced mathematical researches. Although at first Sylow found reading Abel's papers a difficult task, soon he found that Abel had achieved a far deeper understanding of the theory of equations than his published papers indicated. Sylow's first attempts to publish some of Abel's unpublished results that he had found in his papers proved a problem. This was due to the fact that he sent a paper containing these results to Crelle's Journal where the paper was considered for publication by Leopold Kronecker. Now the difficulty arose since Kronecker had already published these results having discovered them himself, and had no wish to have a paper in print which showed that Abel had proved them long before he had. Kronecker, therefore, rejected Sylow's paper but, of course, since Sylow had correctly portrayed Abel's unpublished results, eventually they were indeed attributed to Abel rather than Kronecker.

In 1861 Sylow obtained a scholarship to travel and visited Berlin and Paris. In Paris he attended lectures by Michel Chasles on the theory of conics, by Joseph Liouville on rational mechanics and by Jean-Marie Duhamel on the theory of limits. He says, in the report he wrote at the end of the scholarship, that he also:-

... made myself acquainted with newer works, particularly in the theory of equations.

In Berlin he had useful discussions with Kronecker but was unable to attend courses by Karl Weierstrass who was ill at the time. Bent Birkeland writes in [5] that, as there were no courses being given in Berlin that interested him:-

... he worked instead in the library, studying number theory and the theory of equations. He also got acquainted with Carl Borchardt, the editor of 'Crelle's Journal' ... It is interesting to note that no lectures in algebra or the theory of equations are mentioned from his stay either in Paris or in Berlin.

In 1862 Sylow lectured at the University of Christiania, substituting for Broch who had been elected to serve in the Storting, the Norwegian parliament. In his lectures Sylow explained Abel's and Galois's work on algebraic equations. A summary of these lectures is presented in [4]. It is worth noting that although he had not proved 'Sylow's theorems' at this time (he published them 10 years later) he did pose a question about them. After proving Cauchy's theorem that a group of order divisible by a prime $p$ has a subgroup of order $p$, Sylow asks whether it can be generalised to powers of $p$. These lectures are significant for several reasons, not least that a young student attending them was Sophus Lie. Sylow's lectures were extremely valuable in giving Lie a fundamental appreciation of a topic to which he would make major contributions. Broch was again in the Storting from 1865 to 1868 and he was keen to have Sylow take over his university teaching during this time. However, the school in Fredrikshald refused to give Sylow leave to teach at the university. In 1869 Broch left his chair of pure mathematics, leaving a vacancy that Sylow was well qualified to have filled. However, the University of Christiania did not rate pure mathematics very highly at that time, preferring more practical and applicable topics. Sylow was too theoretical in his approach so he was not appointed.

In 1870-71 Sylow exchanged nine letters with Julius Petersen who, at this time, was working on his doctoral dissertation. Petersen sought Sylow's advice about the main theorem of his dissertation and these letters all deal with this. The article [8] discusses these letters. The two mathematicians exchanged another sixteen letters in 1876-77. Between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete work published under the title Oeuvres complète de Niels Henrik Abel Ⓣ. The motivation for this had come from the Norwegian Academy of Science who applied to the Norwegian Parliament for funding for the project which was quickly granted. This funding allowed Sylow to take leave from his school for four years in order to devote himself to the project. Arild Stubhaug writes in [3]:-

In the course of the work Sylow had an eager discussant in C A Bjerknes, who for his part, worked on the Abel biography that was published in 1880. Bjerknes wanted as much as possible of Abel's early works to come out, not only his great treatises with their exemplary stringency, and perhaps there had been more Abel material than what Holmboe used in his edition of 1839?

Much additional Abel material was found and published in the Sylow/Lie edition which appeared on 9 December 1881. Lie said that most of the work was done by Sylow and commented that Sylow deserved a university position because of his:-

... broad knowledge, his sharp powers of criticism, and his outstanding mathematical work.

However, today Sylow's fame rests on one 10 page paper published in 1872. In this paper Théorèmes sur les groupes de substitutions which Sylow published in Mathematische Annalen Volume 5 (pages 584 to 594) appear the three Sylow theorems although we know that he had already proved his famous theorem by September 1870. Cauchy had already proved that a group whose order is divisible by a prime $p$ has an element of order $p$. Sylow generalised this, proving what is perhaps the most profound result in the theory of finite groups:

If $p^{n}$ is the largest power of the prime $p$ to divide the order of a group $G$ then

1. $G$ has subgroups of order $p^{n}$
2. $G$ has $1 + kp$ such subgroups
3. any two such subgroups are conjugate.

Almost all work on finite groups uses Sylow's theorems. Sylow's original 1872 paper is discussed by Rod Gow in [7] and also by the authors of [6] and the author of [15]. In [11] Winfried Scharlau describes how Sylow was led to his discovery by his study of Galois' work, in particular of Galois' criterion for the solvability of equations of prime degree. The paper [11] explains how Sylow used methods from Galois theory in his proofs.

Sylow became an editor of Acta Mathematica, was elected a member of the Academy of Sciences of Göttingen in 1883, and, in 1894, was awarded an honorary doctorate from the University of Copenhagen.

Lie had a special chair created for Sylow at Christiania University and Sylow taught at the university from 1898. Note that he was 65 years old before he obtained a university post but, remarkably, he was still able to hold this position for 20 years. G A Miller writes [10]:-

Notwithstanding the advanced age at which Sylow entered the university faculty he is said to have filled the position during twenty years with marked success. The duties of his professorship did not seem to be burdensome to him until the last year of his life when he frequently remarked that he felt tired.

Although at the age of 65 he had at last become a university professor, we should note that he did not receive the salary of a professor. At first, he was paid a headmaster's salary which was approximately half the salary of a university professor. Later he received salary increases.

In 1902 Sylow gave the welcoming address at a conference to mark the centenary of Niels Abel's birth. He said (see [2]):-

In the early nineteenth century, applied mathematics had already achieved great triumphs, especially in the fields of astronomy and physics. But just at the same time mathematics ... started to turn its gaze back to the pure and abstract theories. [Gauss and Cauchy] initiated that great movement, which has run through the whole of the previous [19th ] century, and which has reformed mathematics from its foundations at the same time it enriched it with new theories. ... It was in this movement that Niels Abel took such a significant part that he will forever he counted as one of the greatest mathematicians ever.

In 1902 Sylow, in collaboration with Elling Holst, published Abel's correspondence. Further Abel documents had been discovered after the Sylow/Lie book came out in 1881 and, at the 'Third Scandinavian Congress of Mathematicians' which was held in Kristiania in 1913, Sylow discussed this new material.

A version of the introduction to Sylow's discussion is at THIS LINK

In [9] G A Miller quotes Georg Frobenius's opinion of Sylow:-

[Sylow] was known to mathematicians of every civilised country on account of a well-known theorem that bears his name. In 1876 Frobenius remarked that "as every educated person knows the Pythagorean theorem so does every mathematician speak of Abel's theorem and Sylow's theorem".

We must not give the impression that the Sylow theorems and the Abel material were Sylow's only mathematical contributions. He also published a few papers on elliptic functions, particularly on complex multiplication, as well as papers on group theory. Finally we should say a little about Sylow's life outside mathematics. He never married but was a warm person with a nice sense of humour. He was an avid lover of being out of doors and often spent summer vacations in the mountains, usually in Kongsvoll, where he studied plants. Kongsvoll is a mountain station providing food and shelter on the route between Oslo and Trondheim, erected when the route was used by pilgrims visiting the shrine of St Olav in Trondheim.