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| In mathematics, '''Wedderburn's little theorem''' states that every [[finite set|finite]] [[domain (ring theory)|domain]] is a [[field (mathematics)|field]]. In other words, for [[finite ring]]s, there is no distinction between domains, [[skew-field]]s and fields. The theorem is essentially equivalent to saying that the [[Brauer group]] 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, <math>\operatorname{Br}(k) = H^2(k^{\text{al}}/k)</math> coincides with <math>H^1(k^{\text{al}}/k)</math>, which in turn vanishes by Hilbert 90.
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| The [[Artin–Zorn theorem]] generalizes the theorem to [[alternative ring]]s: every finite simple alternative ring is a field.<ref>{{cite book | last=Shult | first=Ernest E. | title=Points and lines. Characterizing the classical geometries | series=Universitext | location=Berlin | publisher=[[Springer-Verlag]] | year=2011 | isbn=978-3-642-15626-7 | zbl=1213.51001 | page=123 }}</ref>
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| == History ==
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| The original proof was given by [[Joseph Wedderburn]] in 1905,<ref name="Lam-2001-p204">Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=204|text=little|p. 204}}</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 {{Harv|Parshall|1983}}, 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.
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| A simplified version of the proof was later given by [[Ernst Witt]].<ref name="Lam-2001-p204"/> Witt's proof is sketched below. Alternatively, the theorem is a consequence of the [[Skolem–Noether theorem]] by the following argument.<ref>Theorem 4.1 in Ch. IV of Milne, class field theory, http://www.jmilne.org/math/CourseNotes/cft.html</ref> Let ''D'' be a finite division algebra with center ''k''. Let [''D'' : ''k''] = ''n''<sup>2</sup> and ''q'' denote the cardinarity of ''k''. Every maximal subfield of ''D'' has ''q<sup>n</sup>'' 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.
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| == Sketch of proof ==
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| Let ''A'' be a finite domain. For each nonzero ''x'' in ''A'', the two maps
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| :<math>a \mapsto ax, a \mapsto xa: A \to A</math>
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| are injective by the cancellation property, and thus, surjective by counting. It follows from the elementary group theory <ref>e.g., Exercise 1.9 in Milne, group theory, http://www.jmilne.org/math/CourseNotes/GT.pdf</ref> that the nonzero elements of ''A'' form a group under multiplication. Thus, ''A'' is a [[Division_ring|skew-field]]. Since the [[Center_(group_theory)|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 ''q<sup>n</sup>''. For each ''x'' in ''A'' that is not in the center, the [[Centralizer and normalizer|centralizer]] ''Z<sub>x</sub>'' of ''x'' has order ''q<sup>d</sup>'' where ''d'' divides ''n'' and is less than ''n''. Viewing ''Z''(''A'')*, ''Z*<sub>x</sub>'' and ''A*'' as groups under multiplication, we can write the [[Conjugacy_class#Conjugacy_class_equation|class equation]]
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| :<math>q^n - 1 = q - 1 + \sum {q^n - 1 \over q^d - 1}</math>
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| where the sum is taken over all representatives ''x'' that is not in ''Z''(''A'') and ''d'' are the numbers discussed above. ''q<sup>n</sup>''−1 and ''q<sup>d</sup>''−1 both admit factorization in terms of [[cyclotomic polynomials]]
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| :<math>\Phi_f(q)</math>.
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| From the polynomial identities
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| :<math>x^n-1 = \prod_{m|n} \Phi_m(x)</math> and <math>x^d-1 = \prod_{m|d} \Phi_m(x)</math>,
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| we set ''x'' = ''q'' to see that
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| :<math>\Phi_n(q)</math> divides both ''q<sup>n</sup>''−1 and <math>{q^n - 1 \over q^d - 1}</math>, | |
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| so by the above class equation <math>\Phi_n(q)</math> must divide ''q''−1, and therefore
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| :<math>|\Phi_n(q)| \leq q-1</math>.<!-- I admit this part is too sketchy; need more details, in particular, for those without knowledge of cyclotomic polynomials. But I don't think any fancy theorems are needed here. -- Taku -->
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| To see that this forces ''n'' to be 1, we will show
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| :<math>|\Phi_n(q)| > q-1</math>
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| for ''n'' > 1 using factorization over the complex numbers. In the polynomial identity
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| :<math>\Phi_n(x) = \prod (x - \zeta)</math>,
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| where ζ runs over the primitive ''n''-th roots of unity, set ''x'' to be ''q'' and then take absolute values
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| :<math>|\Phi_n(q)| = \prod |q - \zeta|</math>.
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| For ''n'' > 1,
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| :<math>|q-\zeta| > |q-1|</math>
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| by looking the location of ''q'', 1, and ζ in the complex plane. Thus
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| :<math>|\Phi_n(q)| > q-1</math>.
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| ==Notes==
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| <references />
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| == References ==
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| * {{cite journal | last = Parshall | first = K. H. | year = 1983 | title = In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen | journal = Archives of International History of Science | volume = 33 | pages = 274–99|authorlink=Karen Parshall}}
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| * {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1= |last2= |first2= |authorlink2= |title=A first course in noncommutative rings |url= |edition=2 |series=Graduate texts in mathematics |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 |id= }}
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| == External links ==
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| *[http://planetmath.org/?op=getobj&from=objects&id=3627 Proof of Wedderburn's Theorem at Planet Math]
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| [[Category:Ring theory]]
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| [[Category:Theorems in abstract algebra]]
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Hi there, I am Felicidad Oquendo. Years in the past we moved to Kansas. Interviewing is how I make a residing and it's some thing I truly appreciate. Climbing is what adore performing.
Feel free to visit my webpage - http://www.january-yjm.com