Bezout identity corollary generalization

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In summary, the writer is asking for a proof using the concept of Bezout's Identity in relation to a corollary and is open to different approaches. The attempted solution involved assuming one factor and then finding additional factors that satisfy the given property. However, the writer has not been able to successfully solve the problem.
  • #1
davon806
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OP warned about not including an attempt at a solution

Homework Statement


Hi,
I have been trying to prove one of the corollaries of the Bezout's Identity in the general form.Unfortunately,I can't figure it out by myself.I hope someone could solve the problem.

If A1,...,Ar are all factors of m and (Ai,Aj) = 1 for all i =/= j,then A1A2...Ar is a factor of m

The writer is asking for a proof using MI on the number of pairs (and associativity to write a product of several intergers as a product of 2 integers, e.g. abcd = (abc)d.However,I am welcome to any different ways of approach.

Homework Equations



Bezout's identity

ax + by = 1

The Attempt at a Solution


I really have no idea on this,and my work was a mess,so I am not going to post it.
 
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  • #2
What have you already tried and where did it break down?

I would start by assuming you only have A1.
m/A1 = X1.
If there is a factor of X1, call it A2, that has the property (A1, A2) = 1, then x*A2=X1, so A1*A2*x = A1*X1 = m.
Build from there.

If at any point your X becomes 1, you have a full set of factors.
 

FAQ: Bezout identity corollary generalization

What is the Bezout identity corollary generalization?

The Bezout identity corollary generalization is a mathematical theorem that extends the Bezout's identity theorem, which states that for two integers a and b, there exist integers x and y such that ax + by = gcd(a,b). The generalization applies this concept to polynomials, stating that for two polynomials f and g, there exist polynomials h and k such that hf + kg = gcd(f,g).

How is the Bezout identity corollary generalization used in mathematics?

This theorem is often used in algebraic geometry and number theory to solve problems involving polynomials. It allows us to find the greatest common divisor of two polynomials and factor them into simpler forms, making it easier to solve equations and prove theorems.

Can the Bezout identity corollary generalization be applied to non-polynomial functions?

No, the generalization only applies to polynomials. However, the concept of finding the greatest common divisor can be applied to other types of functions using different methods.

How does the Bezout identity corollary generalization relate to the Chinese remainder theorem?

The Chinese remainder theorem is a special case of the Bezout identity corollary generalization, where the polynomials f and g are relatively prime. In this case, the solution of hf + kg = gcd(f,g) is unique and can be used to solve simultaneous congruences.

What is the difference between Bezout's identity theorem and the Bezout identity corollary generalization?

Bezout's identity theorem only applies to integers, while the generalization extends this concept to polynomials. Additionally, the generalization allows for multiple solutions, while the original theorem only has one solution. Furthermore, the generalization can be used to solve more complex problems involving polynomials, while Bezout's identity theorem is limited in its applications.

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