Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

Bézout's lemma is a consequence of the Euclidean division defining property, namely that the division by a nonzero integer b has a remainder strictly less than | b |.

A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

The easiest proof of Euclid's lemma uses another lemma called Bézout's identity.

Given a, gcd ( a, n ) = 1, finding x satisfying ax ≡ 1 ( mod n ) is the same as solving ax + ny = 1, which can be done by Bézout's lemma.

An integral domain in which Bézout's identity holds is called a Bézout domain.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

This important property is known as Bézout's identity.

This expression is called Bézout's identity.

Besides finding the greatest common divisor of integers a and b, as the Euclidean algorithm does, it also finds integers x and y ( one of which is typically negative ) that satisfy Bézout's identity

Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves.

The most delicate part of Bézout's theorem and its generalization to the case of k algebraic hypersurfaces in k-dimensional projective space is the procedure of assigning the proper intersection multiplicities.

This field is at least as old as Bézout's theorem.

The intersection of two cubics, which is points ( by Bézout's theorem ), is special in that nine points in general position are contained in a unique cubic, while if they are contained in two cubics they in fact are contained in a pencil ( 1-parameter linear system ) of cubics, whose equations are the projective linear combinations of the equations for the two cubics.

The intersection number is partly motivated by the desire to define intersection to satisfy Bézout's theorem.

In algebra, the polynomial remainder theorem or little Bézout's theorem is an application of polynomial long division.

The degree of a hypersurface F = 0 is the same as the total degree of the homogeneous polynomial F defining it ( granted, in case F has repeated factors, that intersection theory is used to count intersections with multiplicity, as in Bézout's theorem ).

As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).

Two ideals A and B in the commutative ring R are called coprime ( or comaximal ) if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals ( a ) and ( b ) in the ring of integers Z are coprime if and only if a and b are coprime.

The properties of algebraic curves, such as Bézout's theorem, give rise to criteria for showing curves actually are transcendental.

( There are four such points by Bézout's theorem.

The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions.

This repetition ends when b = 0, in which case the variables hold the solution to Bézout's identity: xA + yB

* Online calculator of Bézout's identity.

The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques.

From the modern point of view, Bézout's treatment was rather heuristic, since he did not formulate the precise conditions for the theorem to hold.

:* Two circles never intersect in more than two points in the plane, while Bézout's theorem predicts four.

Some credible sources also name him the founder of the Bézout's identity.

One needs a definition of intersection number in order to state results like Bézout's theorem.

It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity ( 2 for a double point, 3 for a cusp ), in accounting for intersections of curves.

But by Bézout's theorem a cubic and a conic have at most 3 × 2 = 6 points in common, unless they have a common component.

The inner automorphisms form a normal subgroup of Aut ( G ), denoted by Inn ( G ); this is called Goursat's lemma.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

The Yoneda lemma is one of the most famous basic results of category theory ; it describes representable functors in functor categories.

One of these, Itō's lemma, expresses the composite of an Itō process ( or more generally a semimartingale ) dX < sub > t </ sub > with a twice-differentiable function f. In Itō's lemma, the derivative of the composite function depends not only on dX < sub > t </ sub > and the derivative of f but also on the second derivative of f. The dependence on the second derivative is a consequence of the non-zero quadratic variation of the stochastic process, which broadly speaking means that the process can move up and down in a very rough way.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

Their intersection is, which can be shown to be non-context-free by the pumping lemma for context-free languages.

* To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages or a number of other methods, such as Ogden's lemma, Parikh's theorem, or using closure properties.

In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.

One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that

Euclid's classical lemma can be rephrased as " in the ring of integers every irreducible is prime ".

Weak König's lemma is provable in ZF, the system of Zermelo – Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF.

However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.

All the different forms of the same verb constitute a lexeme and the form of the verb that is conventionally used to represent the canonical form of the verb ( one as seen in dictionary entries ) is a lemma.

In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 ( Moore 1982: 168 ).