Among his major accomplishments were the 1940 proof, of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations for algebraic geometry to support that result ( from 1942 to 1946, most intensively ).

On October 1 he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.

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

Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields.

In his last letter to Chevalier and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:

Elliptic curve cryptography ( ECC ) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields.

Specifically, FIPS 186-3 has ten recommended finite fields:

The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.

: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.

The finite fields are classified as follows:

* Any two finite fields with the same number of elements are isomorphic.

This classification justifies using a naming scheme for finite fields that specifies only the order of the field.

See also general linear group over finite fields.

The internal losses are due to absorption and the small but finite losses suffered in the numerous internal reflections due to deviations from the prescribed, cylindrical fiber cross-section and minute imperfections of the core-jacket interface.

But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

Also, no finite field F is algebraically closed, because if a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > n </ sub > are the elements of F, then the polynomial ( x − a < sub > 1 </ sub >)( x − a < sub > 2 </ sub >) ··· ( x − a < sub > n </ sub >) + 1

All numbers which can be obtained from the integers using a finite number of integer additions, subtractions, multiplications, divisions, and taking nth roots ( where n is a positive integer ) are algebraic.

Most AES calculations are done in a special finite field.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

* Incidence algebras of locally finite partially ordered sets are unitary associative algebras considered in combinatorics.

This in turn implies that all finite extensions are algebraic.

The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood.

This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K < sup > sep </ sup >, of degree > 1.

Bessel functions of the first kind, denoted as J < sub > α </ sub >( x ), are solutions of Bessel's differential equation that are finite at the origin ( x = 0 ) for integer α, and diverge as x approaches zero for negative non-integer α.

Procedures for testing hypotheses about probabilities ( using finite samples ) are due to Ramsey ( 1931 ) and de Finetti ( 1931, 1937, 1964, 1970 ).

In this sense they are reminiscent of the oscillatory universe proposed by Richard Chace Tolman: however in Tolman's model the total age of the universe is necessarily finite, while in these models this is not necessarily so.

These languages are exactly all languages that can be decided by a finite state automaton.

The difference between the two definitions is that under the former, finite sets are also considered to be countable, while under the latter definition, they are not considered to be countable.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

More precisely, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other.

This sequence starts with the natural numbers including zero ( finite cardinals ), which are followed by the aleph numbers ( infinite cardinals of well-ordered sets ).

The function F is called prefix-free if there are no two elements p, p ′ in its domain such that p ′ is a proper extension of p. This can be rephrased as: the domain of F is a prefix-free code ( instantaneous code ) on the set of finite binary strings.

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

The key notions in the definition are ( 1 ) that some n is specified at the start, ( 2 ) for any n the computation only takes a finite number of steps, after which the machine produces the desired output and terminates.

Hardwired control units are implemented through use of sequential logic units, featuring a finite number of gates that can generate specific results based on the instructions that were used to invoke those responses.

In fact, all C *- algebras that are finite dimensional as vector spaces are of this form, up to isomorphism.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups.

The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978

The extent of the impulse response is finite, and this would be classified as a fourth-order FIR filter.

It is classified as a wasting asset due to the finite term of the license.

Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:

Coxeter groups were introduced as abstractions of reflection groups, and finite Coxeter groups were classified in 1935.

The simple complex finite dimensional Lie superalgebras were classified by Victor Kac.

classified all quivers of finite type, and also their indecomposable representations.

Other work included that of Felix Klein in computing the invariant rings of finite group actions on ( the binary polyhedral groups, classified by the ADE classification ); these are the coordinate rings of du Val singularities.

Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points.

Not every Jordan algebra is formally real, but classified the finite dimensional formally real Jordan algebras.

Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified.

From 1888 to 1890, Killing essentially classified the complex finite dimensional simple Lie algebras, inventing the notions of a Cartan subalgebra and the Cartan matrix.

Reflection groups over finite fields of characteristic not 2 were classified in.