Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician , best known for his contributions to the theory of elliptic functions , differential equations , number theory , and to group theory . He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions (nowadays known as Padé approximants ), and gave the first full proof for the Cayley–Hamilton theorem . He also lent his name to certain differential-geometric objects in modern mathematical physics , known as Frobenius manifolds .
25-1117: Frobenius is a surname. Notable people with the surname include: Ferdinand Georg Frobenius (1849–1917), mathematician Frobenius algebra Frobenius endomorphism Frobenius inner product Frobenius norm Frobenius method Frobenius group Frobenius theorem (differential topology) Georg Ludwig Frobenius (1566–1645), German publisher Johannes Frobenius (1460–1527), publisher and printer in Basel Hieronymus Frobenius (1501–1563), publisher and printer in Basel, son of Johannes Ambrosius Frobenius (1537–1602), publisher and printer in Basel, son of Hieronymus Leo Frobenius (1873–1938), ethnographer Nikolaj Frobenius (born 1965), Norwegian writer and screenwriter August Sigmund Frobenius (died 1741), German chemist See also [ edit ] Frobenius Orgelbyggeri , Danish organ building firm [REDACTED] Surname list This page lists people with
50-637: A Collegium medico-chirurgicum in 1723; a botanical garden in 1718; and a laboratory in 1753. However, those were later taken over by the University of Berlin . As a French-language institution its publications were in French such as the Histoire de l'Académie royale des sciences et belles lettres de Berlin which was published between 1745 and 1796. A linguistics historian from Princeton University , Hans Aarsleff , notes that before Frederick ascended
75-587: A few topics, the Prussian Academy was the first to teach both sciences and humanities. In 1710, the Academy statute was set, dividing the Academy into two sciences and two humanities classes. This was not changed until 1830, when the physics-mathematics and the philosophy-history classes replaced the four old classes. The reign of King Frederick II of Prussia ("Frederick the Great") saw major changes to
100-523: A sense a dense subset of elements which are accessible to detailed study. Prussian Academy of Sciences The Royal Prussian Academy of Sciences ( German : Königlich-Preußische Akademie der Wissenschaften ) was an academy established in Berlin , Germany on 11 July 1700, four years after the Prussian Academy of Arts , or "Arts Academy," to which "Berlin Academy" may also refer. In
125-462: Is p mod m . From this point of view, the distribution of Frobenius conjugacy classes in Galois groups over Q (or, more generally, Galois groups over any number field) generalizes Dirichlet's classical result about primes in arithmetic progressions. The study of Galois groups of infinite-degree extensions of Q depends crucially on this construction of Frobenius elements, which provides in
150-421: Is a finite Galois extension then to each (positive) prime p which does not ramify in K and to each prime ideal P lying over p in K there is a unique element g of Gal( K / Q ) satisfying the condition g ( x ) = x (mod P ) for all integers x of K . Varying P over p changes g into a conjugate (and every conjugate of g occurs in this way), so the conjugacy class of g in
175-546: Is different from Wikidata All set index articles Ferdinand Georg Frobenius Ferdinand Georg Frobenius was born on 26 October 1849 in Charlottenburg , a suburb of Berlin , from parents Christian Ferdinand Frobenius, a Protestant parson, and Christine Elizabeth Friedrich. He entered the Joachimsthal Gymnasium in 1860 when he was nearly eleven. In 1867, after graduating, he went to
200-425: The University of Göttingen where he began his university studies, but he only studied there for one semester before returning to Berlin, where he attended lectures by Kronecker , Kummer and Karl Weierstrass . He received his doctorate (awarded with distinction) in 1870 supervised by Weierstrass. His thesis was on the solution of differential equations. In 1874, after having taught at secondary school level first at
225-415: The surname Frobenius . If an internal link intending to refer to a specific person led you to this page, you may wish to change that link by adding the person's given name (s) to the link. Retrieved from " https://en.wikipedia.org/w/index.php?title=Frobenius&oldid=1070390346 " Category : Surnames Hidden categories: Articles with short description Short description
250-416: The 18th century, when French was the language of science and culture, it was a French -language institution. Prince-elector Frederick III of Brandenburg , Germany founded the Academy under the name of Kurfürstlich Brandenburgische Societät der Wissenschaften ("Electoral Brandenburg Society of Sciences") upon the advice of Gottfried Wilhelm Leibniz , who was appointed president. Unlike other Academies,
275-508: The Academy under the name of Deutsche Akademie der Wissenschaften zu Berlin (English: German Academy of Sciences at Berlin ) on 1 July 1946. In 1972, it was renamed Akademie der Wissenschaften der DDR or AdW (English: Academy of Sciences of the GDR ). At its height, the AdW had 400 researchers and 24,000 employees in locations across East Germany . Following German Reunification , the Academy
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#1732775358636300-654: The Academy. In 1744, the Nouvelle Société Littéraire and the Society of Sciences were merged into the Königliche Akademie der Wissenschaften ("Royal Academy of Sciences"). An obligation from the new statute were public calls for ideas on unsolved scientific questions with a monetary reward for solutions. The Academy acquired its own research facilities in the 18th century, including an observatory in 1709; an anatomical theater in 1717;
325-470: The Galois group is canonically associated to p . This is called the Frobenius conjugacy class of p and any element of the conjugacy class is called a Frobenius element of p . If we take for K the m th cyclotomic field , whose Galois group over Q is the units modulo m (and thus is abelian, so conjugacy classes become elements), then for p not dividing m the Frobenius class in the Galois group
350-815: The Joachimsthal Gymnasium then at the Sophienrealschule, he was appointed to the University of Berlin as an extraordinary professor of mathematics. Frobenius was only in Berlin a year before he went to Zürich to take up an appointment as an ordinary professor at the Eidgenössische Polytechnikum . For seventeen years, between 1875 and 1892, Frobenius worked in Zürich. It was there that he married, brought up his family, and did much important work in widely differing areas of mathematics. In
375-593: The Prussian Academy was not directly funded out of the state treasury. Frederick granted it the monopoly on producing and selling calendars in Brandenburg, a suggestion from Leibniz. As Frederick was crowned " King in Prussia " in 1701, creating the Kingdom of Prussia , the Academy was renamed Königlich Preußische Sozietät der Wissenschaften ("Royal Prussian Society of Sciences"). While other Academies focused on
400-400: The academy was in a crisis for two decades at mid-century, due to scandals and internal rivalries such as the debates between Newtonianism and Leibnizian views, and the personality conflicts between the philosopher Voltaire and the mathematician Maupertuis . At a higher level, Maupertuis, the director from 1746 to 1759 and a monarchist , argued that the action of individuals was shaped by
425-552: The character of the institution that contained them, and they worked for the glory of the state. By contrast, d'Alembert took a republican rather than monarchical approach and emphasized the international Republic of Letters as the vehicle for scientific advance. By 1789, however, the academy had gained an international repute while making major contributions to German culture and thought. Frederick invited Joseph-Louis Lagrange to succeed Leonhard Euler as director; both were world-class mathematicians. Other intellectuals attracted to
450-540: The character table of the group P S L ( 2 , p ) {\displaystyle PSL(2,p)} of order (1/2)( p − p) for all odd primes p (this group is simple provided p > 3). He also made fundamental contributions to the representation theory of the symmetric and alternating groups . Frobenius introduced a canonical way of turning primes into conjugacy classes in Galois groups over Q . Specifically, if K / Q
475-552: The corresponding committee's members. University departments emanated from some of these businesses after 1945. On 25 November 1915 Albert Einstein presented his field equations of general relativity to the Academy. Under the rule of Nazi Germany from 1933 to 1945, the Academy was subject to the Gleichschaltung , a "Nazification" process that was established to take totalitarian control over various aspects of society. However, compared with other institutions, such as
500-426: The last days of December 1891 Kronecker died and, therefore, his chair in Berlin became vacant. Weierstrass, strongly believing that Frobenius was the right person to keep Berlin in the forefront of mathematics, used his considerable influence to have Frobenius appointed. In 1893 he returned to Berlin, where he was elected to the Prussian Academy of Sciences . Group theory was one of Frobenius' principal interests in
525-653: The philosopher's kingdom were Francesco Algarotti , Jean-Baptiste de Boyer , and Julien Offray de La Mettrie . Immanuel Kant published religious writings in Berlin which would have been censored elsewhere in Europe. Beginning in 1815, research businesses led by Academy committees (such as the Greek-Roman Archeology Committee or the Oriental Committee) were founded at the Academy. They employed mostly scientists to work alongside
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#1732775358636550-472: The second half of his career. One of his first contributions was the proof of the Sylow theorems for abstract groups. Earlier proofs had been for permutation groups . His proof of the first Sylow theorem (on the existence of Sylow groups) is one of those frequently used today. More important was his creation of the theory of group characters and group representations , which are fundamental tools for studying
575-509: The structure of groups. This work led to the notion of Frobenius reciprocity and the definition of what are now called Frobenius groups . A group G is said to be a Frobenius group if there is a subgroup H < G such that In that case, the set together with the identity element of G forms a subgroup which is nilpotent as John G. Thompson showed in 1959. All known proofs of that theorem make use of characters. In his first paper about characters (1896), Frobenius constructed
600-468: The throne in 1740, the academy was overshadowed by similar bodies in London and Paris . Frederick made French the official language and speculative philosophy the most important topic of study. The membership was strong in mathematics and philosophy, and included notable philosophers such as Immanuel Kant , Jean-Baptiste le Rond d'Alembert , Pierre-Louis de Maupertuis , and Etienne de Condillac . However,
625-753: The universities where Jewish employees and members were expelled starting in 1933, Jewish Academy members were not expelled until 1938, following a direct request by the Ministry of Education. The new Academy statute went into effect on 8 June 1939, reorganizing the Academy according to the Nazi leadership principle (the Führerprinzip ). Following World War II , the Soviet Military Administration in Germany , or SMAD , reorganized
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