Wedderburn's little theorem
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Wedderburn's little theorem states that every
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. In other words, for
finite ring In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite grou ...
s, there is no distinction between domains,
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s and fields. The
Artin–Zorn theorem In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin. ...
generalizes the theorem to
alternative ring In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is o ...
s: every finite alternative division ring is a field.


History

The original proof was given by
Joseph Wedderburn Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fie ...
in 1905,Lam (2001),
p. 204 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the ''Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Repo ...
/ref> who went on to prove it two other ways. Another proof was given by
Leonard Eugene Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ...
shortly after Wedderburn's original proof, and Dickson acknowledged Wedderburn's priority. However, as noted in , Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs appeared only after he had read Dickson's correct proof. On this basis, Parshall argues that Dickson should be credited with the first correct proof. A simplified version of the proof was later given by
Ernst Witt Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...
. Witt's proof is sketched below. Alternatively, the theorem is a consequence of the
Skolem–Noether theorem In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in ...
by the following argument. Let D be a finite
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fie ...
with
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
k. Let :kn^ and q denote the cardinality of k. Every maximal subfield of D has q^ elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of D in our case) cannot be a union of conjugates of a proper subgroup; hence, n = 1. A later "
group-theoretic In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as g ...
" proof was given by
Ted Kaczynski Theodore John Kaczynski ( ; born May 22, 1942), also known as the Unabomber (), is an American domestic terrorist and former mathematics professor. Between 1978 and 1995, Kaczynski killed three people and injured 23 others in a nationwide ...
in 1964. This proof, Kaczynski's first published piece of mathematical writing, was a short, two-page note which also acknowledged the earlier historical proofs.


Relationship to the Brauer group of a finite field

The theorem is essentially equivalent to saying that the
Brauer group Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik ...
of a finite field is trivial. In fact, this characterization immediately yields a proof of the theorem as follows: let ''k'' be a finite field. Since the
Herbrand quotient In mathematics, the Herbrand quotient is a quotient of orders of Group cohomology, cohomology groups of a cyclic group. It was invented by Jacques Herbrand. It has an important application in class field theory. Definition If ''G'' is a finite cyc ...
vanishes by finiteness, \operatorname(k) = H^2(k^/k) coincides with H^1(k^/k), which in turn vanishes by Hilbert 90.


Proof

Let ''A'' be a finite domain. For each nonzero ''x'' in ''A'', the two maps :a \mapsto ax, a \mapsto xa: A \to A are injective by the
cancellation property In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An ...
, and thus, surjective by counting. It follows from the elementary group theorye.g., Exercise 1.9 in Milne, group theory, http://www.jmilne.org/math/CourseNotes/GT.pdf that the nonzero elements of A form a group under multiplication. Thus, A is a
skew-field In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
. To prove that every finite skew-field is a field, we use strong induction on the size of the skew-field. Thus, let A be a skew-field, and assume that all skew-fields that are proper subsets of A are fields. Since the
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
Z(A) of A is a field, A is a vector space over Z(A) with finite dimension n. Our objective is then to show n = 1. If q is the order of Z(A), then A has order ^. Note that because Z(A) contains the distinct elements 0 and 1, q>1. For each x in A that is not in the center, the
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
_ of x is clearly a skew-field and thus a field, by the induction hypothesis, and because _ can be viewed as a vector space over Z(A) and A can be viewed as a vector space over _, we have that _ has order ^ where d divides n and is less than n. Viewing ^, A^, and the ^_ as groups under multiplication, we can write the
class equation In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
:q^n - 1 = q - 1 + \sum where the sum is taken over the conjugacy classes not contained within ^, and the d are defined so that for each conjugacy class, the order of ^_ for any x in the class is ^ - 1. ^ - 1 and q^ - 1 both admit
polynomial factorization In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field (mathematics), field or in the integers as the product of irreducible polynomial, irreducible ...
in terms of
cyclotomic polynomials In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primit ...
:\Phi_f(q). In the polynomial identities :x^n-1 = \prod_ \Phi_m(x) and x^d-1 = \prod_ \Phi_m(x), we set x = q. Because each d is a proper divisor of n, :\Phi_n(q) divides both ^ - 1 and each , so by the above class equation \Phi_n(q) must divide q - 1, and therefore :, \Phi_n(q), \leq q-1. To see that this forces n to be 1, we will show :, \Phi_n(q), > q-1 for n>1 using factorization over the complex numbers. In the polynomial identity :\Phi_n(x) = \prod (x - \zeta), where \zeta runs over the primitive n-th roots of unity, set x to be q and then take absolute values :, \Phi_n(q), = \prod , q - \zeta, . For n>1, we see that for each primitive n-th root of unity \zeta, :, q-\zeta, > , q-1, because of the location of q, 1, and \zeta in the complex plane. Thus :, \Phi_n(q), > q-1.


Notes


References

* * {{cite book , last1=Lam , first1=Tsit-Yuen , title=A first course in noncommutative rings , edition=2 , series=
Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard s ...
, volume=131 , year=2001 , publisher=Springer , isbn=0-387-95183-0


External links


Proof of Wedderburn's Theorem at Planet Math
*
Mizar system The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used in ...
proof: http://mizar.org/version/current/html/weddwitt.html#T38 Theorems in ring theory