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77-435: Peter Ludvig Meidell Sylow ( IPA: [ˈsyːlɔv] ) (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory . Sylow processed and further developed the innovative works of mathematicians Niels Henrik Abel and Évariste Galois in algebra . Sylow theorems and p-groups , known as Sylow subgroups , are fundamental in finite groups . By profession, Sylow

154-677: A Cro-Magnon population that arrived in Europe about 45,000 years ago, Neolithic farmers who migrated from Anatolia during the Neolithic Revolution 9,000 years ago, and Yamnaya steppe pastoralists who expanded into Europe from the Pontic–Caspian steppe in the context of Indo-European migrations 5000 years ago. The Norwegian population is typical of the Northern European population with Haplogroup I1 being

231-520: A Scandinavian meeting of naturalists in 1860 in Copenhagen, where he presented a solid interpretation of a strange equation-theoretic treatise by Abel, edited only in fragments. In 1861 Sylow obtained a scholarship for studies in Paris and Berlin. 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

308-637: A union with Denmark . In 1814, following Denmark–Norway's defeat in the Napoleonic Wars , Norway entered a union with Sweden and adopted a new constitution . Rising nationalism throughout the 19th century led to a 1905 referendum granting Norway independence . Although Norway remained officially neutral in World War I , the country was unofficially allied with the Entente powers . In World War II , Norway proclaimed its neutrality, but

385-608: A captain in the cavalry, and later he served as the head of the Ministry of the Army between 1848 and 1854. Initially, his father was aware of his son's talent in Mathematics , so he encouraged him to work independently. From home, Sylow learned a sense of duty and hard work, but was also taught to be modest and although this was done with the best of intentions, it would become an obstacle for him later in life since it meant that he

462-745: A chance to get back to Norway. There are very few of them left there today. According to recent genetic analysis, both mtDNA ( mitochondrial DNA ) and Y-chromosome polymorphisms showed a noticeable genetic affinity between the Norwegian population and other ethnic groups in Northern and Central Europe , particularly with the Germans. This is due to a history of at least a thousand years of large-scale migration both in and out of Norway. Norwegians, like most Europeans, largely descend from three distinct lineages: Mesolithic hunter-gatherers , descended from

539-761: A common culture and speak the Norwegian language . Norwegians are descended from the Norse of the Early Middle Ages who formed a unified Kingdom of Norway in the 9th century. During the Viking Age , Norwegians and other Norse peoples conquered, settled and ruled parts of the British Isles , the Faroe Islands , Iceland and Greenland . Norwegians are closely related to other descendants of

616-632: A large membership, although many people do not remain observant as adults. A majority of both ethnic Norwegians and Sámi are nominally Christian, but not necessarily observant. In Norway as of 2018, 70% of the population are members of the Lutheran Church , though only 47.1% answered "Yes" to the question "Do you believe in God?" in a 2018 European Values Study . The Norwegians are and have been referred to by other terms as well. Some of them include: Theory of equations Before Galois, there

693-426: A lecturer, while schools were poorer for having him as a teacher. During his studies, Sylow had become interested in the work of Niels Henrik Abel , and especially in an unfinished work on equation theory that had been left behind. However, it was only at Hartvig Nissen School (1856–58) that he began to research that work more deeply, in part thanks to Ole Jacob Broch, who was the school's pure mathematics teacher at

770-525: A long reign that came to an end when Sylow was finally appointed as a professor of mathematics at the University of Oslo in 1898, and despite already being 65 when he obtained a university post, he was still able to hold this position for 20 years, until 1918, when he died at the age of 85. His rare talent in mathematics revealed itself immediately upon his arrival at the university, to which he brought knowledge far beyond elementary mathematics. At first, he

847-448: A lot. However, it was only when Sylow began to lecture about Abel's and Galois's work on algebraic equations in 1862, that he began to further develop their innovative works, especially those related to group theory, and in fact, by the end of that year, Sylow had already proved foundational results in group theory, which are now known as Sylow theorems and p-groups , known as Sylow subgroups , which are now basic terms in group theory. He

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924-405: A particularly good school teacher, since he was interested in the advanced areas of mathematics and had thus little enthusiasm for teaching at lower levels. Moreover, he also 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

1001-577: A solution for the quartic equations , discovered by his student Lodovico Ferrari . In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations. The case of higher degrees remained open until the 19th century, when Paolo Ruffini gave an incomplete proof in 1799 that some fifth degree equations cannot be solved in radicals followed by Niels Henrik Abel 's complete proof in 1824 (now known as

1078-600: A year doing the mathematical lectures at the Christiania University during Broch's absence abroad, during which he also began to treat and lecture Galois' group theory . But instead, his career simply stopped. When Broch again became an MP in the Storting from 1865 to 1868, he was keen to have Sylow take over the teaching of his university during this time, but the school in Fredrikshald in which Saylow

1155-1098: Is a North Germanic language with approximately 5 million speakers, of whom most are located in Norway . There are also some speakers of Norwegian in Denmark , Sweden , Germany , Britain , Spain , Canada , and the United States , where the largest community of speakers exists, with 55,311 speakers as of 2000; approximately half of the speakers live in Minnesota (8,060), California (5,865), Washington (5,460), New York (4,200), and Wisconsin (3,520). As of 2006, in Canada, there are 7,710 Norwegian speakers , of whom 3,420 reside in British Columbia , 1,360 in Alberta , and 1,145 in Ontario . Norwegian culture

1232-513: Is also a commonly revered holiday. It marks midsummer and the beginning of summer vacation, and is often celebrated by lighting bonfires the evening before. In Northern areas of Norway, this day has 24 hours of light, while southern areas have only 17.5 hours. The conversion of Norway to Christianity from Norse paganism began in 1000. By the middle of the 11th century, Christianity had become well-established in Norway and had become dominant by

1309-478: Is closely linked to the country's history and geography . The unique Norwegian farm culture , sustained to this day, has resulted not only from scarce resources and a harsh climate but also from ancient property laws . In the 18th century, it brought about a strong romantic nationalistic movement, which is still visible in the Norwegian language and media . In the 19th century, Norwegian culture blossomed as efforts continued to achieve an independent identity in

1386-584: Is now called "Sylow's theorem", which he had known since 1862. Jordan was astonished and somewhat skeptical, but shortly afterwards, he wrote enthusiastically from Sweden , and he helped Sylow to get that 10-page thesis published that same year in 1872. That thesis made Sylow a well-known European mathematician. In 1868 he was elected into the Norwegian Academy of Science and Letters ( Det Norske Widenskaps-Akademi ). From 1870 to 1871, Sylow exchanged nine letters with Julius Petersen who, at this time,

1463-571: The Constitution of Norway . Many people wear bunad (traditional costumes) and most participate in or watch the Norwegian Constitution Day parade that day, consisting mostly of children, through the cities and towns. The national romanticist author Henrik Wergeland was the founder of the 17 May parade. Common Christian holidays are also celebrated, the most important being Christmas (called Jul in Norway after

1540-733: The United States , Canada , Australia , New Zealand , South Africa and the United Kingdom. Norwegians are traditionally Lutheran since the Reformation in Denmark–Norway and Holstein which made Lutheranism the only legal religion in the country, however large portions of the population are now either non-practicing, atheist or agnostic . Towards the end of the 3rd millennium BC, Proto-Indo-European –speaking Battle-Axe peoples migrated to Norway bringing domesticated horses , agriculture , cattle and wheel technology to

1617-410: The 19th century, "theory of equations" was almost synonymous with "algebra". For a long time, the main problem was to find the solutions of a single non-linear polynomial equation in a single unknown . The fact that a complex solution always exists is the fundamental theorem of algebra , which was proved only at the beginning of the 19th century and does not have a purely algebraic proof. Nevertheless,

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1694-465: The 2000 U.S. Census, three million Americans consider Norwegian to be their sole or primary ancestry. It is estimated that as many as a further 1.5 million more are of partial Norwegian ancestry. Norwegian Americans represent 2–3% of the non-Hispanic Euro-American population in the U.S. They mostly live in both the Upper Midwest and Pacific Northwest . As early as 1814, a party of Norwegians

1771-534: The 8th century and onwards, Norwegian and Danish Vikings also settled in Normandy , most famously those led by Rollo ; some of their Norman descendants would later expand to England, Sicily , and other Mediterranean islands. Apart from Britain and Ireland, Norwegian Vikings established settlements in largely uninhabited regions. The first known permanent Norwegian settler in Iceland was Ingólfur Arnarson . In

1848-549: The Canadian Arctic. Norwegian immigration to Canada lasted from the mid-1880s until 1930, although Norwegians were already working in Canada as early as 1814. It can be divided into three periods of roughly fifteen years each. In the first, to about 1900, thousands of Norwegians homesteaded on the Canadian prairies. In the second, from 1900 to 1914, there was a further heavy influx of Norwegians immigrating to Canada from

1925-582: The Dutch merchant fleet and navy came to consist of Norwegians and Danes. Most took Dutch names , leaving no trace of Norwegian names in the later Dutch population. The emigration to the Netherlands was so devastating to the homelands that the Danish-Norwegian king issued penalties of death for emigration, but repeatedly had to issue amnesties for those willing to return, announced by posters in

2002-864: The Norsemen such as Danes , Swedes , Icelanders and the Faroe Islanders , as well as groups such as the Scots whose nation they significantly settled and left a lasting impact in, particularly the Northern Isles ( Orkney and Shetland ). The Norwegian language, with its two official standard forms, more specifically Bokmål and Nynorsk , is part of the larger Scandinavian dialect continuum of generally mutually intelligible languages in Scandinavia . Norwegian people and their descendants are found in migrant communities worldwide, notably in

2079-618: 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 in Fredrikshald for four years in order to devote himself to the project. Sylow wanted as much as possible of Abel's early works to come out, not just his great treatises with their exemplary stringency, and thus, he used this opportunity to dig up more of his early works, and in fact, there

2156-553: The United States ( Michigan , Minnesota , North Dakota , South Dakota , and Montana ) show Scandinavian (which includes Norwegian) ancestry proportions among European descent (white) persons of 10 to 20%. Similarly, Norwegian ancestry has been found to account for about 25% of ancestry of the population of the Shetland Islands and Danish-Norwegian ancestry has been found to account for about 25% of ancestry of

2233-500: The United States because of poor economic conditions in the US, and 18,790 from Norway. In the third, from 1919 to 1930, 21,874 people came directly from Norway, with the peak year in 1927, when 5,103 Norwegians arrived, spurred by severe depression at home. They came with limited means, many leaving dole queues. From 1825 to 1900 some 500,000 Norwegians landed at Ville du Quebec in Canada (and other Canadian ports) for travelling through Canada

2310-494: The United States. As a result of this, many of the Norwegians remaining in Norway were religiously moderate; subsequently, church attendance declined throughout the 20th century, as reflected by 78% of the population stating that religion is unimportant in a Gallup poll and low weekly church attendance, at 2%, particularly when compared to that of North Dakota , the state in which Norwegians constitute approximately 30.4% of

2387-482: The United States. In 1850, the year after Great Britain repealed its restrictive Navigation Acts in Canada, more emigrating Norwegians sailed the shorter route to the Ville de Québec ( Quebec City ) in Canada, to make their way to US cities like Chicago , Milwaukee , and Green Bay by steamship . For example, in the 1850s, 28,640 arrived at Quebec, Canada, en route to the US, and 8,351 at New York directly. According to

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2464-507: The areas of literature, art and music. Norway's culinary traditions show the influence of long seafaring and farming traditions with salmon (fresh and cured), herring (pickled or marinated), trout , codfish and other seafood balanced by cheeses, dairy products and excellent breads (predominantly dark/darker). Lefse is a common Norwegian potato flatbread, common around Christmas. For renowned Norwegian dishes, see lutefisk , smalahove , pinnekjøtt , Krotekake and fårikål . Along with

2541-582: The classical music of romantic composer Edvard Grieg and the modern music of Arne Nordheim , Norwegian black metal has become something of an export article in recent years. Norway's classical performers include Leif Ove Andsnes , one of the world's more famous pianists, and Truls Mørk , an outstanding cellist. The jazz scene in Norway is also thriving. Jan Garbarek , Mari Boine , Arild Andersen , and Bugge Wesseltoft are internationally recognised while Paal Nilssen-Love , Supersilent , Jaga Jazzist and Wibutee are becoming world-class artists of

2618-569: The country's generous social safety net with an aging population , and preserving economic competitiveness. Norwegian or Norse Vikings raided and settled in Shetland , Orkney , Ireland, Scotland, and northern England. In the United Kingdom, many names for places ending in - kirk , - ness , - thorpe , - toft and - by are likely Norse in origin. In 947, a new wave of Norwegian Vikings appeared in England when Erik Bloodaxe captured York . In

2695-576: The country's official religion, and its central administrative functions were carried out by the Royal Ministry of Government Administration, Reform and Church Affairs until 2017. The Lutheran Church is still mentioned in the constitution , for example, the King is still required to profess a Lutheran faith. When baptised, children are registered in the Church of Norway 's member register, leading to

2772-412: The editor there, Leopold 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 did not accept that Abel had preceded him, and therefore, he rejected Sylow's paper, but even though the article was rejected, posterity has proved Sylow right. Sylow showcased his discoveries at

2849-529: The high school mathematics teacher's examination ( Realkankidat , Norwish to Real candidate) with excellent grades. He completed his graduation in 1856, but since no university post was available, he taught for two years at Hartvig Nissen School , an independent girls' school in the Uranienborg district of Christiania , which had been founded by Hartvig Nissen and Ole Jacob Broch . His years there came during Broch's most energetic university period, and it

2926-429: The main concern of the algebraists was to solve in terms of radicals, that is to express the solutions by a formula which is built with the four operations of arithmetics and with nth roots . This was done up to degree four during the 16th century. Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations . Gerolamo Cardano published them in his 1545 book Ars Magna , together with

3003-539: The middle of the 12th century. The Norwegians were Catholics until the Danish king Christian III of Denmark forced them to convert to Lutheranism and established a state-governed church. The church undertook a program to convert the Sámi in the 16th and 17th century, with the program being largely successful. In the 19th century, emigration from Norway for political and religious motives began and Lutheranism spread to

3080-493: The most common Y-haplogroup, at about 37,3%. Norwegians also show the characteristic R1a genes of the paternal ancestorship at 17.9% to 30.8%. Such large frequencies of R1a have been found only in East Europe and India. R1b gene showing paternal descent is also widespread at 25.9% to 30.8%. Norwegian genetic ancestry also exists in many locations where Norwegians immigrated. In particular, several northern states in

3157-465: The one who will stand as the permanent. Sylow 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

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3234-544: The pagan and early Viking winter solstice) and Easter (Påske). In Norway, the Santa (called Nissen ) comes at Christmas Eve , the 24 December, with the presents, not the morning after as in many English speaking countries. He usually comes late in the evening, after the Christmas dinner many children consider long, boring and unnecessary. Jonsok ( St. John 's Passing), or St. Hans ( St. John 's Day), i.e. 24 June,

3311-401: The population of Greenland . Y-Chromosome DNA ( Y-DNA ) represents the male lineage, The Norwegian Y-chromosome pool may be summarized as follows where haplogroups R1 & I comprise generally more than 85% of the total chromosomes. Mitochondrial DNA ( mtDNA ) represents the female lineage, Haplogroup H represent about 40% of the Norwegian mitochondrial DNA lineages Norwegian

3388-466: The population. Of all U.S. states, North Dakota has the lowest percentage of non-religious people and the largest number of churches per capita. It weekly church attendance is at 43%. In Norway the Church of Norway and state are not entirely separated. An act approved in 2016 created the Church of Norway as an independent legal entity, effective from 1 January 2017. The Church of Norway was previously

3465-484: The prospect of becoming a lecturer at a university, where he had absolutely belonged from the first moment, but the disfavor of the times left him unnoticed in his native land despite his name being already widely known outside Norway. As a result, Sylow spent a whole 40 years, from 1858 to 1898, holding the modest position of head teacher in mathematics and science at the Frederiksborg Latin School,

3542-454: The region. During the Viking Age , Harald Fairhair unified the Norse petty kingdoms after being victorious at the Battle of Hafrsfjord in the 880s. Two centuries of Viking expansion tapered off following the decline of Norse paganism with the adoption of Christianity in the 11th century. During The Black Death , approximately 60% of the population died and in 1397 Norway entered

3619-546: The same time. According to the 2011 Census , 452,705 Canadians reported Norwegian ancestry (Norwegian-Canadians). As of 2011, there were 3,710 Norwegian-born Australians, and 23,037 Norwegians of Australian descent. In the 19th century a community known as the Kola Norwegians settled in the environs of the Russian city of Murmansk . They have suffered persecution under Joseph Stalin and after 1992 were offered

3696-527: The shrine of St Olav in Trondheim. Sylow died on 7 September 1918, at the age of 85, in Christiania, Norway. Norwegians b. ^ There are millions of Britons of Scandinavian ancestry and ethnicity, though mixed with others. Norwegians ( Norwegian : Nordmenn ) are an ethnic group and nation native to Norway , where they form the vast majority of the population. They share

3773-424: The solvability of equations of prime degree. The paper explains how Sylow used methods from Galois theory in his proofs. Besides the thesis of 1872, Sylow's main work was the new edition of Abel's collected writings which he procured in association with former student Sophus Lie on a public basis, when in 1873, Sylow and Lie were commissioned to provide a new edition of Niels Henrik Abel's collected works, paid for by

3850-416: The state. The preparations for the publication of this work took eight years, from 1873 to 1881, during which he had only partial leave from his teaching work, being on leave from school for four years. Sylow and Lie prepared an edition of Abel's complete work published under the title Oeuvres complète de Niels Henrik Abel (French for: Complete Works of Niels Henrik Abel). The motivation for this had come from

3927-515: The streets of Amsterdam. Increasingly, Dutchmen who search their genealogical roots turn to Norway. Many Norwegians who emigrated to the Netherlands, and often were employed in the Dutch merchant fleet, emigrated further to the many Dutch colonies such as New Amsterdam (New York). Many Norwegians emigrated to the US between the 1850s and the 1920s. The descendants of these people are known as Norwegian Americans. Many Norwegian settlers traveled to and through Canada and Canadian ports while immigrating to

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4004-537: The theory of equations. In the following year, in 1862, Sylow lectured at the University of Christiania as a substitute for Professor Ole Jacob 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, and in doing so he became one of the first in Europe to lecture on Évariste Galois 's works. Among his listeners

4081-403: The theory of finite groups. Almost all work on finite groups uses Sylow's theorems. When the famous French mathematician Camille Jordan published the standard work Théorie des Substitutions in 1870, Sylow was familiar with most of what was written there and more. When Jordan visited Christiania in 1872, Sophus Lie took him on an excursion to Frognerseteren with Sylow, who described to him what

4158-480: The theory of limits. He also used this scholarship to make himself acquainted with newer works, particularly in the theory of equations. In Berlin, Sylow had useful discussions with Kronecker, but was unable to attend courses by Karl Weierstrass who was ill at the time, and since there were no other courses being given in Berlin that interested him, Sylow instead decided to work in the library, studying number theory and

4235-652: The time. It was Broch who 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, he managed to struggle through them and soon found that Abel had achieved a far deeper understanding of the theory of equations than what he had managed to write in his published papers. Some of Sylow's first attempts to publish some of Abel's unpublished results that he had found in his papers proved to be unsuccessful. For instance, he sent one of these papers to Crelle's Journal in Berlin , but

4312-562: The year 874 he settled in Reykjavík . After his expulsion from Iceland Erik the Red discovered Greenland, a name he chose in hope of attracting Icelandic settlers. Viking settlements were established in the sheltered fjords of the southern and western coast. Erik's relative Leif Eriksson later discovered North America. During the 17th and 18th centuries, many Norwegians emigrated to the Netherlands, particularly Amsterdam . The Netherlands

4389-414: The younger generation. Norway has a strong folk music tradition which remains popular to this day. Among the most prominent folk musicians are Hardanger fiddlers Andrea Een , Olav Jørgen Hegge , Vidar Lande and Annbjørg Lien , violinist Susanne Lundeng , and vocalists Agnes Buen Garnås , Kirsten Bråten Berg and Odd Nordstoga . Norwegians celebrate their national day on 17 May, dedicated to

4466-696: Was Broch who introduced Sylow to Carl Gustav Jacob Jacobi 's fundamental work on elliptic functions , among other things. In 1858, Sylow moved to the town of Fredrikshald (now called Halden ) in Ostfold county, where he taught at Frederiksborg Latin School as the Head Teacher in Mathematics and Science, a modest position that he held for a whole 40 years, from 1858 to 1898. Although Sylow would have made an outstanding university lecturer, he did not make

4543-544: Was a teacher at the Frederiksborg Latin School for 40 years from 1858 to 1898, and then a professor at the University of Oslo for 20 years from 1898 to 1918. Despite the isolation in Frederiksborg, Sylow was an active member of the mathematical world. He wrote a total of approximately 25 mathematical and biographical works, corresponded with many of the leading mathematicians of the time, and

4620-540: Was a teacher refused to give him leave to teach at the Christiania university, and they received support from the ministry. Broch left his chair as professor of pure mathematics in 1869, thus leaving a vacancy that Sylow was well qualified to have filled, and in fact, everyone expected Sylow to take over his professorship in pure mathematics. However, the University of Christiania did not rate pure mathematics very highly at that time, preferring more practical, useful, down-to-earth mathematics with more applicable topics, and Sylow

4697-464: Was also true of the many Americans of Norwegian heritage who immigrated to Canada from the US with "Canada Fever" seeking homesteads and new economic opportunities. By 1921 one-third of all Norwegians in Canada had been born in the US. These new Canadians became British subjects in Canada, and part of the British Empire. Canadian citizenship, as a status distinct from that of a British subject,

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4774-711: Was an able co-editor of Acta Mathematica from the journal's start in 1882. He was also elected into the Norwegian Academy of Science and Letters in 1868, a corresponding member of the Academy of Sciences in Göttingen and the University of Copenhagen awarded him an honorary doctorate in 1894. Ludvig Sylow was born in Kristiania (now Oslo ) on 12 December 1832 to later minister and customs treasurer Thomas Edvard von Westen Sylow (1792–1875) and Magdalene Cecilie Cathrine Mejdell (1806–98). His father had been an officer and

4851-640: Was brought to Canada to build a winter road from York Factory on Hudson Bay to the infant Red River settlement at the site of present-day Winnipeg, Manitoba , Canada. Norway House is one of the oldest trading posts and Native-Canadian missions in the Canadian West. Willard Ferdinand Wentzel served the North West Company of Canada in the Athabasca and Mackenzie regions and accompanied Sir John Franklin on his overland expedition in 1819–20 to

4928-465: Was created on 1 January 1947, with Canada being the first Commonwealth country to create their own citizenship. Prior to that date, Canadians were British subjects and Canada's nationality law closely mirrored that of the United Kingdom. On 1 January 1947, Canadian citizenship was conferred on most British subjects connected with Canada. Unlike the US, Canada was part of the British Empire and most Norwegians would have become Canadians and British subjects at

5005-446: Was happy to spend many years in a more lowly position than he should have had. Sylow attended Christiania Cathedral School, graduating in 1850 after taking the examen artium . He then became a student at the University of Oslo where he began his studies of natural sciences. In 1853, the University of Oslo awarded him the Crown Prince's gold medal ( Kronprinsens gullmedalje ) for a Mathematics subject about Gnomonics . In 1856 he took

5082-504: Was held in Kristiania in 1913, Sylow discussed this new material. In addition to the Sylow theorems and the Abel material, Sylow also published a few papers on elliptic functions, particularly on complex multiplication, as well as papers on group theory. In 1883 Sylow became an editor of Acta Mathematica , was elected a member of the Academy of Sciences of Göttingen and, in 1894, the University of Copenhagen awarded him an honorary doctorate. A couple of times in his youth, Sylow briefly had

5159-404: Was much more additional Abel material published in the Sylow/Lie edition which appeared on 9 December 1881, than what Bjerknes used on his Abel biography of 1880. 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

5236-402: Was no clear distinction between the "theory of equations" and "algebra". Since then algebra has been dramatically enlarged to include many new subareas, and the theory of algebraic equations receives much less attention. Thus, the term "theory of equations" is mainly used in the context of the history of mathematics , to avoid confusion between old and new meanings of "algebra". Until the end of

5313-407: Was nonetheless occupied for five years by Nazi Germany (1940–45). In 1949, neutrality was abandoned and Norway became a member of NATO . Discovery of oil and gas in adjacent waters in the late 1960s boosted Norway's economy but in referendums held in 1972 and 1994 , Norway rejected joining the EU . Key domestic issues include integration of a fast-growing immigrant population , maintaining

5390-416: Was paid a headmaster's salary, which was approximately half the salary of a university professor, but he later received salary increases. At the centenary of Abel's birth in 1902, Sylow gave the welcoming address at a conference to mark the centenary of his birth, giving a characterization of his great predecessor, who was hailed by all the famous mathematicians of the various countries who had met together as

5467-466: Was published in Mathematische Annalen Volume 5 (pages 584 to 594). This paper has the three Sylow theorems, which prove foundational results in group theory. Sylow had already proved this in 1862, but only published it in 1872, and by then, Augustin-Louis Cauchy had already proved that a group whose order is divisible by a prime. Winfried Scharlau described how Sylow was led to his discovery by his study of Galois' work, in particular of Galois' criterion for

5544-533: Was the second-most popular destination for Norwegian emigrants after Denmark. Loosely estimated, some 10% of the population may have emigrated, in a period when the entire Norwegian population consisted of some 800,000 people. The Norwegians left with the Dutch trade ships that when in Norway traded for timber, hides, herring, and stockfish (dried codfish). Young women took employment as maids in Amsterdam, while young men took employment as sailors. Large parts of

5621-485: Was the shortest corridor to the United States' central states. In spite of efforts by the Government of Canada to retain these immigrants for Canada, very few remained because of Canada's somewhat restrictive land policies at that time and negative stories being told about Canada from U.S. land agents deterring Norwegians from going to Canada. Not until the 1880s did Norwegians accept Canada as a land of opportunity. This

5698-461: Was the young Sophus Lie , who would later create a strange new science on the basis of these ideas, the theory of continuous symmetry . Lie once commented that Sylow deserved a university position because of his "broad knowledge, his sharp powers of criticism, and his outstanding mathematical work". And for a time, it seemed that the university would finally bet on him since he had received a scholarship trip to Berlin and Paris in 1861, and then spent

5775-483: Was thus one of the first mathematicians to penetrate Galois' group theory. However, it was not until 1872, 10 years later, that Sylow published his most important discoveries in group theory in Alfred Clebsch's journal ( Math. Ann. ), in a small treatise of ten pages called Theorémes sur les groupes de substitutions , in which Sylow generalizes his discoveries and proves what is perhaps the most profound result in

5852-817: Was too theoretical in his approach so he was not appointed. The professor of applied mathematics, Carl Anton Bjerknes , was instead pressured to move into Broch's position, so that Cato Guldberg could take over the applied mathematics. Sylow was known to mathematicians of every civilized country on account of a well-known theorem that bears his name. Just as every educated person knows the Pythagorean theorem so does every mathematician speak of Abel's theorem and Sylow's theorem. Ferdinand Georg Frobenius Since few contemporary mathematicians were as deeply familiar with Abel's work as Sylow was, Professor Carl Anton Bjerknes advised him to study Évariste Galois 's works about group theory , in which Abel had also contributed

5929-450: Was working on his doctoral dissertation. Petersen sought Sylow's advice about the main theorem of his dissertation and all of these letters deal are about this subject. The two mathematicians exchanged another sixteen letters a few years later, in 1876 and 1877. However, Sylow's most well-known written work rests on his 10-page thesis published in 1872 called Théorèmes sur les groupes de substitutions (Theorems on substitution groups), which

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