Everything about Ferdinand Georg Frobenius totally explained
Ferdinand Georg Frobenius (
October 26,
1849 –
August 3,
1917) was a
German mathematician, best-known for his contributions to the theory of
differential equations and to
group theory. He also gave the first full proof for the
Cayley-Hamilton theorem.
Frobenius was born in
Charlottenburg, a suburb of
Berlin, and was educated at the
University of Berlin. His thesis was on the solution of differential equations, under the direction of
Weierstrass. After its completion in
1870, he taught in Berlin for a few years before receiving an appointment at the Polytechnicum in
Zurich (now
ETH Zurich). In
1893 he returned to Berlin, where he was elected to the
Prussian Academy of Sciences.
Contributions to group theory
Group theory was one of Frobenius' principal interests in the second half of his career. One of his first notable 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 the structure of groups. This work led to the notion of
Frobenius reciprocity and the definition of what are now called
Frobenius groups. He also
made fundamental contributions to the
character theory of the symmetric groups.
Contributions to number theory
Frobenius introduced a canonical way of turning primes into
conjugacy classes in
Galois groups over
Q. Specifically, if
K/
Q is a finite Galois extension then to each (positive) prime
p which doesn't
ramify in
K and to each prime ideal
P lying over
p in
K there's a unique element
g of Gal(
K/
Q) satisfying the condition
g(
x) =
xp (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 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
mth
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 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 a sense a dense subset of elements which are accessible to detailed study.
Further Information
Get more info on 'Ferdinand Georg Frobenius'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://ferdinand_georg_frobenius.totallyexplained.com">Ferdinand Georg Frobenius Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
