In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Wedderburn's little theorem states that every
finite
Finite may refer to:
* 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 marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
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 ...
; thus, every finite domain is a field. 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 (or, occasionally, a sfield), 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 multiplicativ ...
s and fields.
The
Artin–Zorn theorem generalizes the theorem to
alternative rings: 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 fi ...
in 1905,
[Lam (2001), p. 204/ref> who went on to prove the theorem in 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 rem ...
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 f ...
. 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 ...
by the following argument. Let 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 . Let and denote the cardinality of . Every maximal subfield of has elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of in our case) cannot be a union of conjugates of a proper subgroup; hence, .
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 ...
" proof was given by Ted Kaczynski
Theodore John Kaczynski ( ; May 22, 1942 – June 10, 2023), also known as the Unabomber ( ), was an American mathematician and domestic terrorist. He was a mathematics prodigy, but abandoned his academic career in 1969 to pursue a reclusi ...
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
In mathematics, the Brauer group of a field ''K'' is an abelian group whose elements are Morita equivalence classes of central simple algebras over ''K'', with addition given by the tensor product of algebras. It was defined by the algebraist ...
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 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 cyclic group acting o ...
vanishes by finiteness, coincides with , which in turn vanishes by Hilbert 90.
The triviality of the Brauer group can also be obtained by direct computation, as follows. Let and let be a finite extension of degree so that Then is a cyclic group of order and the standard method of computing cohomology of finite cyclic groups shows that
where the norm map is given by
Taking to be a generator of the cyclic group we find that has order and therefore it must be a generator of . This implies that is surjective, and therefore is trivial.
Proof
Let ''A'' be a finite domain. For each nonzero ''x'' in ''A'', the two maps
:
are injective by the cancellation property
In mathematics, the notion of cancellativity (or ''cancellability'') is a generalization of the notion of invertibility.
An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M ...
, and thus, surjective by counting. It follows from elementary group theory[e.g., Exercise 1-9 in Milne, group theory, http://www.jmilne.org/math/CourseNotes/GT.pdf] that the nonzero elements of form a group under multiplication. Thus, is a division ring
In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
.
Since the center of is a field, is a vector space over with finite dimension . Our objective is then to show . If is the order of , then has order . Note that because contains the distinct elements and , . For each in 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 \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
of is a vector space over , hence it has order where is less than . Viewing , , and as groups under multiplication, we can write the class equation
In mathematics, especially group theory, two elements a and b of a Group (mathematics), 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 c ...
:
where the sum is taken over the conjugacy classes not contained within , and the are defined so that for each conjugacy class, the order of for any in the class is . In particular, the fact that is a subgroup of implies that divides , whence divides by elementary algebra.
and both admit polynomial factorization
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same doma ...
in terms of cyclotomic polynomial
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 prim ...
s . The cyclotomic polynomials on are in