Great video. I have been getting legitimately frustrated and angry because I'm doing online classes now due to the ongoing corona lockdown, and my prof. did not thoroughly cover the details regarding the structure of polynomial factor rings or provide any lecture material, but basically told us to use our textbook to work out the computational details. It's very frustrating because textbooks usually don't go very deep into the details of their computations or examples, and there are very few video resources available for abstract algebra and other mid-to-high level mathematics.
I'm glad you found it helpful. This video series was created when I was forced to teach the class online, and I figured I'd put them up for everyone exactly for people in your situation.
That is one of the things that I find fascinating about math in general, and abstract algebra in particular. You can be studying something from an entirely different area of math, and find that it is intimately tied together with a topic that at first seems completely unrelated.
Hi; i take this video as my main reference on the subject, so well explained that i keep coming back from time to time; i would like to ask if it could be right to say there is a hint on the absortion property of ideals in it? i mean an absortion property such that operations on the quotient turn back to the ideal as the zero-class(?), in the sense of an additive (group) zero class (?) thanks a lot once again!!
The 0 for a ring is the additive identity. It is confusing because the elements of our factor ring are huge sets of polynomials, but in this case the additive identity is the ring of all multiples of x^2+1. The true additive identity is , but to understand the simplification trick, it is easier to just consider the "defining polynomial" x^2+1 as being 0.
A rather dumb question probably: what are some applications of a polinomyal quotient ring (beside the homomorphism' theorem.that states that a (given) ring homomorphism is isomorphic to the quotient ring defined by the ideal? Thanks
I'm honestly not the best person to answer that question; I typically teach lower level courses than this one, so I'm not up on all the details (especially after a couple of years since I went through this). I can't really point to a specific use of this particular topic. However, a lot of this is building its way up to Galois theory, which has many clever applications.
@@patrickjones1510 Hi; your video prompted me to get into the first stages of study of quotient rings, which had always been quite alien to me and quite hard to pin down, so i really thank you; perhaps you could tell some reference book on the subject? Greetings
Dear Patrick ,....I have a question from this topic could you please help me,....my question is: what is ring ℤₚ[x]) / ⟨x²⟩,,...I mean to what ring it looks like which is easy to see,....is it isomorphic to ℤₚ² ?I have strong intuttion that it is isomorphic to ℤₚ² but can't prove it,.....if it is not isomorphic to ℤₚ²,then to which ring it is isomorphic?
Great video. I have been getting legitimately frustrated and angry because I'm doing online classes now due to the ongoing corona lockdown, and my prof. did not thoroughly cover the details regarding the structure of polynomial factor rings or provide any lecture material, but basically told us to use our textbook to work out the computational details. It's very frustrating because textbooks usually don't go very deep into the details of their computations or examples, and there are very few video resources available for abstract algebra and other mid-to-high level mathematics.
I'm glad you found it helpful. This video series was created when I was forced to teach the class online, and I figured I'd put them up for everyone exactly for people in your situation.
@@patrickjones1510 I'm in the same situation, now I have to learn everything online. Your videos are really helping. Thanks!!
Thank you great video explaining how quotients in polynomial rings actually works.
Thanks a lot, I'm starting to study rings on my own and this helped me a lot with the examples in my book.
I had been stuck here for a while. Thank you so much for clearing my doubt! I have my algebra exam tomorrow wish me luck !
Amazing professor..... Brilliant work!! Thanks a lot
I was blown away when i realized about it being isomorphic to complex numbers. Somehow i didn't notice x^2=-1 (which means x=i) until you metioned it)
That is one of the things that I find fascinating about math in general, and abstract algebra in particular. You can be studying something from an entirely different area of math, and find that it is intimately tied together with a topic that at first seems completely unrelated.
Hi; i take this video as my main reference on the subject, so well explained that i keep coming back from time to time; i would like to ask if it could be right to say there is a hint on the absortion property of ideals in it? i mean an absortion property such that operations on the quotient turn back to the ideal as the zero-class(?), in the sense of an additive (group) zero class (?) thanks a lot once again!!
What a great video! Thanks
Excellent explanation!
Thank you soo much for cleaning my dought's 😊 I need this
I have a question, because I'm not sure if I understand this.
Why can we say that x^2+1 is effectively 0?
The 0 for a ring is the additive identity. It is confusing because the elements of our factor ring are huge sets of polynomials, but in this case the additive identity is the ring of all multiples of x^2+1. The true additive identity is , but to understand the simplification trick, it is easier to just consider the "defining polynomial" x^2+1 as being 0.
good, now i understand why it can be treat as complex numbers
Amazing! Thank you
A rather dumb question probably: what are some applications of a polinomyal quotient ring (beside the homomorphism' theorem.that states that a (given) ring homomorphism is isomorphic to the quotient ring defined by the ideal? Thanks
I'm honestly not the best person to answer that question; I typically teach lower level courses than this one, so I'm not up on all the details (especially after a couple of years since I went through this). I can't really point to a specific use of this particular topic. However, a lot of this is building its way up to Galois theory, which has many clever applications.
@@patrickjones1510 thank you! Great video; greetings
@@patrickjones1510 Hi; your video prompted me to get into the first stages of study of quotient rings, which had always been quite alien to me and quite hard to pin down, so i really thank you; perhaps you could tell some reference book on the subject? Greetings
Dear Patrick ,....I have a question from this topic could you please help me,....my question is: what is ring ℤₚ[x]) / ⟨x²⟩,,...I mean to what ring it looks like which is easy to see,....is it isomorphic to ℤₚ² ?I have strong intuttion that it is isomorphic to ℤₚ² but can't prove it,.....if it is not isomorphic to ℤₚ²,then to which ring it is isomorphic?
Since the factor x² does not break into distinct factors i cannot use Chinese remainder theorem here
Thank you
you are the best!
Well done. Thanks.
ty🖤
awesome tnx
Very useful sir.
omg Thank u so much!!
You're welcome. I'm glad you found it helpful.
Now prove there’s an inverse 😢