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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 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 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 ...
" 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, \operatorname(K) = H^2(K^/K) coincides with H^1(K^/K), which in turn vanishes by Hilbert 90. The triviality of the Brauer group can also be obtained by direct computation, as follows. Let , K, = q, and let L/K be a finite extension of degree n, so that , L, =q^n. Then \mathrm(L/K) is a cyclic group of order n, and the standard method of computing cohomology of finite cyclic groups shows that H^2(L/K) = K^/N_(L^), where the norm map N_:L^ \to K^ is given by N_(\alpha) = \prod_ \sigma(\alpha) = \alpha \cdot \alpha^ \cdot \alpha^ \cdots \alpha^ = \alpha^. Taking \alpha to be a generator of the cyclic group L^, we find that N_(\alpha) has order q-1, and therefore it must be a generator of K^. This implies that N_ is surjective, and therefore H^(L/K) is trivial.


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 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 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
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 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 \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...
_ of x is a vector space over Z(A), hence it has order ^ where d is less than n. Viewing ^, A^, 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 ...
: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. In particular, the fact that ^_ is a subgroup of A^ implies that q^d-1 divides q^n-1, whence d divides n by elementary algebra. ^ - 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 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 \Phi_f(q). The cyclotomic polynomials on \Q are in \Z /math>, and satisfy the identities :x^n-1 = \prod_ \Phi_m(x) and x^d-1 = \prod_ \Phi_m(x). Since each d is a proper divisor of n, :\Phi_n(x) divides both ^ - 1 and each in \Z /math>, thus by the class equation above, \Phi_n(q) must divide q - 1, and therefore by taking the norms, :, \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) () 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 size (with va ...
, 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 ...
proof: http://mizar.org/version/current/html/weddwitt.html#T38 Theorems in ring theory