In
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
* ...
is a
field. In other words, for
finite rings, 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 generalizes the theorem to
alternative rings: every finite alternative division ring is a field.
History
The original proof was given by
Joseph Wedderburn in 1905,
[Lam (2001), ]p. 204 P. is an abbreviation or acronym that may refer to:
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* Paris Herbarium, at the ''Muséum national d'histoire naturelle''
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* The ''Pacific Repo ...
/ref> who went on to prove it two other ways. Another proof was given by Leonard Eugene Dickson 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. Witt's proof is sketched below. Alternatively, the theorem is a consequence of the Skolem–Noether theorem by the following argument. Let be a finite division algebra 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" 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 vanishes by finiteness, coincides with , which in turn vanishes by Hilbert 90.
Proof
Let ''A'' be a finite domain. For each nonzero ''x'' in ''A'', the two maps
:
are injective by the cancellation property, and thus, surjective by counting. It follows from the 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 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 be a skew-field, and assume that all skew-fields that are proper subsets of are fields. 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 of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
of is clearly a skew-field and thus a field, by the induction hypothesis, and because can be viewed as a vector space over and can be viewed as a vector space over , we have that has order where divides and is less than . Viewing , , and the as groups under multiplication, we can write the class equation
:
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 . 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 dom ...
in terms of cyclotomic polynomials
:
In the polynomial identities
: and ,
we set . Because each is a proper divisor of ,
: divides both and each ,
so by the above class equation must divide , and therefore
:
To see that this forces to be , we will show
:
for using factorization over the complex numbers. In the polynomial identity
:
where runs over the primitive -th roots of unity, set to be and then take absolute values
:
For , we see that for each primitive -th root of unity ,
:
because of the location of , , and in the complex plane. Thus
:
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 ...
, volume=131 , year=2001 , publisher=Springer , isbn=0-387-95183-0
External links
Proof of Wedderburn's Theorem at Planet Math
* Mizar system proof: http://mizar.org/version/current/html/weddwitt.html#T38
Theorems in ring theory