Mathematical theorems

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Amitsur–Levitzki theorem

In algebra, the Amitsur–Levitzki theorem states that the algebra of n by n matrices satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n. More details Android, Windows

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Hadamard's inequality

Another inequality is called the Hermite–Hadamard inequality. In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||. Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then | det ( N ) | ≤ ∏ i = 1 n ∥ v i ∥ . {\displaystyle |\det(N)|\leq \prod _{i=1}^{n}\|v_{i}\|.} If the n vectors are linearly independent, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. More details Android, Windows

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Eisenstein's criterion

In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases to prove irreducibility with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. More details Android, Windows

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Schur's lemma

For other uses, see Schur's lemma (disambiguation). In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N and φ is a self-map. The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which is due to Jacques Dixmier. More details Android, Windows

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Euler's four-square identity

In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. More details Android, Windows

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Hasse's theorem on elliptic curves

Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If N is the number of points on the elliptic curve E over a finite field with q elements, then Helmut Hasse's result states that | N − ( q + 1 ) | ≤ 2 q . {\displaystyle |N-(q+1)|\leq 2{\sqrt {q}}.} That is, the interpretation is that N differs from q + 1, the number of points of the projective line over the same field, by an 'error term' that is the sum of two complex numbers, each of absolute value √q. This result had originally been conjectured by Emil Artin in his thesis. It was proven by Hasse in 1933, with the proof published in a series of papers in 1936. Hasse's theorem is equivalent to the determination of the absolute value of the roots of the local zeta-function of E. In this form it can be seen to be the analogue of the Riemann hypothesis for the function field associated with the elliptic curve. More details Android, Windows

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Frobenius theorem

In mathematics, more specifically in abstract algebra, the Frobenius theorem, proved by Ferdinand Georg Frobenius in 1877, characterizes the finite-dimensional associative division algebras over the real numbers. According to the theorem, every such algebra is isomorphic to one of the following: R (the real numbers) C (the complex numbers) H (the quaternions). These algebras have dimensions 1, 2, and 4, respectively. Of these three algebras, R and C are commutative, but H is not. More details Android, Windows

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Stone's representation theorem for Boolean algebras

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone (1936), and thus named in his honor. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. More details Android, Windows

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Sylow theorems

In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Ludwig Sylow (1872) that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups. For a prime number p, a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group G is a maximal p-subgroup of G, i.e. a subgroup of G that is a p-group (so that the order of every group element is a power of p) that is not a proper subgroup of any other p-subgroup of G. The set of all Sylow p-subgroups for a given prime p is sometimes written Sylp(G). The Sylow theorems assert a partial converse to Lagrange's theorem. While Lagrange's theorem states that for any finite group G the order (number of elements) of every subgroup of G divides the order of G, the Sylow theorems state that for every prime factor p of the order of a finite group G, there exists a Sylow p-subgroup of G. The order of a Sylow p-subgroup of a finite group G is pn, where n is the multiplicity of p in the order of G, and every subgroup of order pn is a Sylow p-subgroup of G. The Sylow p-subgroups of a group (for a given prime p) are conjugate to each other. The number of Sylow p-subgroups of a group for a given prime p is congruent to 1 mod p. More details Android, Windows

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Primitive element theorem

In field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions. It says that a finite extension is simple if and only if there are only finitely many intermediate fields. In particular, finite separable extensions are simple. More details Android, Windows

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Poincaré–Birkhoff–Witt theorem

For the Poincaré–Birkhoff fixed-point theorem, see Poincaré–Birkhoff theorem. In mathematics, more specifically in abstract algebra, in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poincaré, Garrett Birkhoff, and Ernst Witt. The terms PBW type theorem and PBW theorem may also refer to various analogues of the original theorem, comparing a filtered algebra to its associated graded algebra, in particular in the area of quantum groups. More details Android, Windows

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Newton's identities

In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. More details Android, Windows

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Nakayama lemma

In mathematics, more specifically modern algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in Nakayama (1951), although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951). In the commutative case, the lemma is a simple consequence of a generalized form of the Cayley–Hamilton theorem, an observation made by Michael Atiyah (1969). The special case of the noncommutative version of the lemma for right ideals appears in Nathan Jacobson (1945), and so the noncommutative Nakayama lemma is sometimes known as the Jacobson–Azumaya theorem. The latter has various applications in the theory of Jacobson radicals. More details Android, Windows

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Modularity theorem

In mathematics, the modularity theorem (formerly called the Taniyama–Shimura–Weil conjecture and several related names) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem. Later, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended Wiles' techniques to prove the full modularity theorem in 2001. The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, Freitas, Le Hung & Siksek (2015) proved that elliptic curves defined over real quadratic fields are modular. More details Android, Windows

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Maschke's theorem

In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. If (V, ρ) is a finite-dimensional representation of a finite group G over a field of characteristic zero, and U is an invariant subspace of V, then the theorem claims that U admits an invariant direct complement W; in other words, the representation (V, ρ) is completely reducible. More generally, the theorem holds for fields of positive characteristic p, such as the finite fields, if the prime p does not divide the order of G. More details Android, Windows

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Cayley's theorem

For the number of labeled trees in graph theory, see Cayley's formula. In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G. This can be understood as an example of the group action of G on the elements of G. A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G). Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups". The regular action used in the standard proof of Cayley's theorem does not produce the representation of G in a minimal-order permutation group. For example, S 3 {\displaystyle S_{3}} , itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of S 6 {\displaystyle S_{6}} (a group of order 720). The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult. More details Android, Windows

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Muirhead's inequality

In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. More details Android, Windows