Abstract Algebra | k[x] is a PID

[@newwaveinfantry8362]

6:05 - Why is the ideal generated by alpha equal to k[x] and not just k? Is it because we're working in the context of k[x].

[@vikramsundara5065]

multiply the element x with 1 in the ideal

[@adelalomar6478]

In Case 2: Do you mean r(x) = alpha and not p(x) = alpha.Or I misunderstood something. But Thanks for the proof

[@carl3260]

5:44 Yep, he means remainder r(x) is a constant (or non-zero deg-0 poly)

[@poonamrane9553]

right. but how does the proceed in that case?@@carl3260

[@jaewoolee5958]

Could we have used the theorem for irreducible polynomial for the proof? Since quotient ring of ring over principal ideal generated by prime polynomials is field, principal ideal generated by prime polynomial is also a maximal one. Therefore, no more than one element could be used to generate an ideal thus ring is a PID.

[@gulnazshalgumbayeva8655]

Hi I am from Kazakhstan ;) Thanks for your videos, very helpful.

[@danielgrunkin8705]

What about the case when I = (0)? In this case I guess we cannot require that deg(p(x)) >= 1 since that would mean that I /= (0).

[@keepitplainsimple1466]

Please don't release so many videos at once. This won't help you grow your youtube channel. Instead, keep some in stock and post your videos regularly thus managing the influx of Viewers. Awesome video love it

[@ProfOmarMath]

I think his channel is growing just fine.

[@MichaelPennMath]

Thanks for the comment, but my overall goals are not just to grow this channel. I have to balance the videos I make for my courses and the ones I make for a general audience. I have no choice but to release the videos for my courses at the moment.

[@ProfOmarMath]

@@MichaelPennMath You've got me thinking about creating videos for my fall courses on here. Since I'm still on sabbatical it'll be much easier to do now rather than during the hectic post-Corona post-sabbatical transition.

[@keepitplainsimple1466]

Actually @MICHAELPENN I can understand your point of view and this is the reason why I am saying this. Now you're getting a massive influx of subs but when you go back to normal life this subscriber base will dwindle. It would really benefit you if you store those videos and release them at regular intervals. Your videos are awesome and I am a Maths Enthusiast myself, of 15 years and am thinking of taking up maths for my life. ☺

[@MichaelPennMath]

I see, at the moment I have ~20 unpublished videos ready to upload. Once the semester is over I will be able to release 1video/day for the foreseeable future.

[@AbirInsights]

Good morning

[@MichaelPennMath]

Same to you