Bézout's identity

The lemma of Bézout states in number theory (after Étienne Bézout ( 1730-1783 ) ), that the greatest common divisor of two integers, and of which at least one is not equal, can be represented as a linear combination of and with integer coefficients:

If and are relatively prime, then there are so

Applies.

The coefficients and can be calculated using the extended Euclidean algorithm efficiently.

The lemma can be generalized to more than two integers: Are integers, then there are integer coefficients with

More generally, the lemma of Bézout applicable in all principal ideal ring, even in a non- commutative; for the precise statements see there.

Evidence

The proof of the lemma is based on the possibility of division with remainder, thus it can be easily transferred to Euclidean rings. Among all the numbers there are certainly those that are positive and. Be the smallest number among them. Since both divides, also shares. Is is also and proved the statement. In the event we show that is also a divisor of and is. Division with remainder provides us with a representation of the form, wherein. If one uses for the presentation and solves the equation after on, you get. Be due to the minimality of need, then, is a divisor of. Similarly, is that a divisor of, and thus applies. Previously, we had already seen that a divisor of is. So true.

Principal ideals

If one uses the concept of the ideal of the ring theory, it must be understood that the main ideals and contained in the principal ideal are. So also the ideal is contained in. One can formulate the lemma of Bézout also so that for the ring ( or more generally for Euclidean rings) applies

Principal ideal rings are rings in which every ideal is a principal ideal. There are about elements and the ring is always an element, so that the ideal is the principal ideal. then on the one hand and a common divisor of, and on the other hand, a linear combination of and. Therefore applicable in principal ideal, so to speak, by definition, the lemma of Bézout if you as the look of the item and.

Conclusions

The lemma of Bézout is for mathematics and especially for the number theory of fundamental importance. Thus, the lemma of Euclid can thus eg derived which has the uniqueness of prime factorization result. For linear Diophantine equations Bézout 's lemma gives a criterion for the solvability.