Abstract Algebra | Polynomial Rings

We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R[x] is as well.
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[@Batman-vh9so]

Honestly best math channel on KZbin! It'd be great if you could also do a series on algebraic and differential topology.

[@michellejingdong]

Really nice and helpful channel! Thank you a lot!

[@realnameverified416]

I didn't know you could give money on comments like that.

[@ashimchakraborty2908]

@#Michael Penn, sir why did you take p(x)=x instead of p(x)=1? What is the point i mean p(x)q(x)=1 when p(x),q(x) are constant polynomial with p(x) and q(x) not equal to 0. Reply please sir

[@MGSchmahl]

The purpose was to prove that there exists a polynomial with no multiplicative inverse. p(x)=1 is trivially its own inverse, and any other 0-degree polynomial p(x)=c has an inverse p'(x)=1/c. Therefore, to provide a counterexample, we have to look to polynomials of degree at least 1, the simplest of which is p(x)=x.

[@ashimchakraborty2908]

@@MGSchmahl Thanks

[@nailabenali7488]

Can you do all the results about degre and roots of a polynomial? Also the fundamental theorem of algebra? And maybe show why a polynomial can't be considered as a funtion ( I mean not the same as his polynomial function) like in (Z/nZ)? Thank you for the great work!!

[@MichaelPennMath]

I have several more polynomial videos waiting to be uploaded. I'll also think about some of those other topics as well.

[@mrlimemil]

As part of these polynomial videos are you planning on doing a video the Heegner numbers? I think your videos on these topics have been great so far and look forward to any new videos you put out.

[@el_ginga]

Thanks Mr Zuckerberg