Abstract Algebra | Introduction to Euclidean Domains

We give the definition of a Euclidean domain, provide some examples including the Gaussian Integers Z[i], and prove that every Euclidean domain is a principal ideal domain (PID).

[nrrgrdn]

3:40 you say conjugation and multiplication commutes. Don't you mean conjugation distributes over multiplication? conj(a*b) = conj(a)*conj(b)

[Hjhtijnkeable]

Sir, I think the given domain and range of the valuation function needs a little inclusion.. According to the definition, what can we say about the valuation image of the zero polynomial of K[x]? And the image of 0 in the domain of Gaussian Integers?? If the domain be non-zero and the range be non-negative Integers, then everything fits.. btw loved the way you interpret Mathematics..

[frederikbosman]

how can you take beta's inverse if its not necessarily invertible in Z[i]?

[behzat]

but inverse of beta might not be in Z[i], how is it okay to start with alpha*beta^(-1) ?

[studiesplusdotlk]

It's a great explanation sir.Thank you

[maxdickens]

I think the domain of N should be D\{0} instead of D.
Because the degree of zero polynomial (i.e. f(x) = 0) SHOULD BE minus infinity. But in your definition, it can't be smaller than zero, which is not good.

[vtk]

Could you should start a series of lessons in mathematics please (such as:Lifting the exponent lemma; ord and primitive Roots;ect. It would be of great help

[SuperSilver]

Your last video has had me stumped on trying to find a direct method towards computing a solution that differential equation, I think it was f(f'(x)) = x. I wrote it in that form since the inverse notation doesn't look so nice in these comments (f'(x) = f^(-1)(x), was the actual problem). I was trying so many different ways to formulate solutions, and I wanted to run something by you. Do you think a power series solution could for work for this problem? The solution you found actually has a pretty nice power series representation if you expand about x =1, and I was wondering if that could work for a problem like this. We can formulate a compositional power series out of f(f'(x)) and try and match coefficients to both sides. The matching wouldn't be that difficult, its the computation of the coefficients that I think would prove tricky.

[Alex-Eldridge]

Does the norm function need to be defined at zero? From what I can tell, the degree of the zero polynomial is either undefined or negative infinity, so I'm not sure how the degree function works as a norm on all of F[x]. It doesn't seem like you used the norm of zero anywhere in the video.

[SurendraSingh-jkyg]

sir can you please prove bezout's theorem as I am preparing for PRMO. I t will help me a lot.
PLEASE 🙏🙏