Abstract Algebra | Writing the gcd of polynomials as a combination.

We prove that the greatest common divisor of any two polynomials over a field can be written as polynomial combinations of the original polynomials. This is Bezout's identity for polynomials.
Please Subscribe: kzbin.info
Personal Website: www.michael-penn.net
Randolph College Math: www.randolphcollege.edu/mathematics/
Research Gate profile: www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en

[@tortoisefanfan]

Dear Prof Micjael Penn: The 72st and 73st course seem to be identical.

[@Tucxy]

Great argument for the uniqueness of a monic polynomial of minimal degree, awesome video

[@jaewoolee5958]

Michael thanks for the lectures. At @11:55 we set r(x) = 0 since remainder belongs in S so must have at least degree of n but also must have lower degree than d(x). Does that mean we can assume the 0 polynomial ring to have multiple degrees at the same time?

[@Maria-yx4se]

do these results hold in n variables

[@xCorvus7x]

9:47 Does the quotient polynomial q(x) have a negative degree, or why is the degree of r(x) not bound by the degree of r(x)q(x)?

[@MichaelPennMath]

All of those details are a result of the division algorithm for polynomials: kzbin.info/www/bejne/lYusfpSYq816oNk

[@xCorvus7x]

@@MichaelPennMath Ups, sorry. Thanks for the link.

[@stevenstat]

When you show that there is a unique monic polynomial in S, you prove uniqueness. But did you ever prove that there IS a monic polynomial of smallest degree? Maybe all of the polynomials in S of smallest degree all have leading coefficient >1.

[@Jkfgjfgjfkjg]

True. However, note that by the well-ordering axiom of Z, there exists a polynomial of minimal degree, and since F is a field, you can divide the leading coefficient by its inverse in F to make it monic.

[@Jkfgjfgjfkjg]

Nice presentation. You should consider writing a book.

[@tucantipack]

♥️♥️♥️♥️♥️

[@CDChester]

Legit thought you wrote "Writing the God of Polynomials" and I was like wow clickbait level maximum.

[@blackpenredpen]

C. D. Chester That would have been so awesome. I would totally want to know the god of polynomials!

[@MichaelPennMath]

Maybe the Taylor series for e^x is the god of polynomials?

[@CDChester]

@@MichaelPennMath probably the most utilized

[@catalinchirita5637]

This is way too hard for me

[@hassan.aladawy]

You probably just needed to know some prerequisites, nothing is too hard!