However, by Wedderburn's little theorem all finite division rings are commutative and therefore finite fields.

Wedderburn's little theorem states that the Brauer group of a finite field is trivial, so that every finite division ring is a finite field.

* Wedderburn's little theorem states that finite domains are fields.

* Wedderburn's little theorem

A finite domain is automatically a finite field by Wedderburn's little theorem.

* Wedderburn's little theorem

# REDIRECT Wedderburn's little theorem

A consequence of the Jacobson density theorem is Wedderburn's theorem ; namely that any right artinian simple ring is isomorphic to a full matrix ring of n by n matrices over a division ring for some n. This can also be established as a corollary of the Artin – Wedderburn theorem.

For finite projective spaces of geometric dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF ( q ), whose order ( that is, number of elements ) is q ( a prime power ).

However, due to Wedderburn's theorem, which states that all finite division rings are fields, all finite Desarguesian planes are Pappian.

We obtain an equivalence relation on CSAs over K by the Artin – Wedderburn theorem ( Wedderburn's part, in fact ), to express any CSA as a M ( n, D ) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying M ( m, D ) with M ( n, D ) for all integers m and n at least 1, we get the Brauer equivalence and the Brauer classes.

:* K is a finite field ( Wedderburn's theorem );

Wedderburn's theorem characterizes simple rings with a unit and a minimal left ideal.

Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.

In an unexpected decision, Lord Kames stated that ' we sit here to enforce right not to enforce wrong ' and the court emphatically rejected Wedderburn's appeal, ruling that ‘ the dominion assumed over this Negro, under the law of Jamaica, being unjust, could not be supported in this country to any extent: That, therefore, the defender had no right to the Negro ’ s service for any space of time, nor to send him out of the country against his consent: That the Negro was likewise protected under the act 1701, c. 6.

This is Joseph Wedderburn's original result.

When Lord Bute was prime minister, Wedderburn used to go on errands for him, and it is to Wedderburn's credit that he first suggested to the premier the propriety of granting Samuel Johnson a pension.

The resignation of Pitt on the question of Catholic emancipation ( 1801 ) put an end to Wedderburn's tenure of the Lord Chancellorship, for, much to his surprise, no place was found for him in Addington's cabinet. Pitt's friends believed he had ben guilty of treachery over the Emancipation issue ; and even the King, who used Loughborough as a spy in Cabinet, later commented that his death removed " the gretest knave in the Kingdom ".

In Wedderburn's character ambition banished all rectitude of principle, but the love of money for money's sake was not among his faults.

Although this work contained no explicit mention of slavery, it does suggest Wedderburn's future path in subversive and radical political action.

It concluded that Wedderburn's expression of concern about possible corruption in relation to the original decision by Liverpool Council had no significant effect on Minister Beamer's decision to reject the revised LEP, which could be justified on legitimate planning grounds.

Conversely, Wedderburn's counsel argued that commercial interests, which underpinned Scotland's prosperity, should prevail.

After the case, Knight became a free man, and married his sweetheart Annie Thompson, a woman who had also been in Wedderburn's service and had been sacked following the revelation of her relations with Knight.

Its first known appearance is in Wedderburn's Complaynt of Scotland ( 1548 ) under the name " The frog came to the myl dur ", though this in Scots rather than English.

The causeway on Wedderburn Road, which provides the main link to Campbelltown, is known to flood during heavy rain, increasing Wedderburn's isolation.

Fermat's little theorem states that all prime numbers have the above property.

# REDIRECT Fermat's little theorem

For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

* A lemma is a " helping theorem ", a proposition with little applicability except that it forms part of the proof of a larger theorem.

* A corollary is a proposition that follows with little or no proof from one other theorem or definition.

This can be used to prove Fermat's little theorem and its generalization, Euler's theorem.

* Leonhard Euler produces the first published proof of Fermat's " little theorem ".

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

Fermat's little theorem states that if p is prime and a is coprime to p, then a < sup > p − 1 </ sup > − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides a < sup > x − 1 </ sup > − 1, then x is called a Fermat pseudoprime to base a.

It is not a prime, since it equals 11 · 31, but it satisfies Fermat's little theorem: 2 < sup > 340 </ sup > ≡ 1 ( mod 341 ) and thus passes

Fermat's little theorem states that if p is a prime number, then for any integer a, the number a < sup > p </ sup > − a is an integer multiple of p. In the notation of modular arithmetic, this says

If a is not divisible by p, Fermat's little theorem is equivalent to the statement that a < sup > p − 1 </ sup > − 1 is an integer multiple of p:

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory.

It is called the " little theorem " to distinguish it from Fermat's last theorem.

( There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.

This is a special case of Fermat's little theorem.