So glad to find your channel. I just bought an abstract algebra book and was looking through a proof in the section "Irreducible Polynomials Are Maximal Ideals" and thought I'd type that "section title" into UA-cam to see what would come up...and here you are. I'm looking forward to watching more of your stuff.
Technically K[x,y] is isomorphic to (K[x])[y], in the sense that the ring of polyals in two variables x and y, is isomorphic to the one only having one variable but the coefficients into K[x]. The problem with that is that while K is a field, we know that K[x] is not. So that's a nice question. But yes, you can prove that every ideal I = (p) where p is an irreducible element inside a Ring is maximal, then the result which involves the quotient R/M, that you can see on the left side of the blackboard, is true without making any assumptions on the qualities R must have. So yes, that should estend onto the multi variable case
So glad to find your channel. I just bought an abstract algebra book and was looking through a proof in the section "Irreducible Polynomials Are Maximal Ideals" and thought I'd type that "section title" into UA-cam to see what would come up...and here you are. I'm looking forward to watching more of your stuff.
Brother you are doing great work..Love from India 🇮🇳🇮🇳🇮🇳
Me too ,from India. Sir, greetings from India
5:00 is the cleanest edit
You can try to prove it yourself and skip ahead to 8:47 to watch the applications.
Does this proof extend to the multi variable case ?
Technically K[x,y] is isomorphic to (K[x])[y], in the sense that the ring of polyals in two variables x and y, is isomorphic to the one only having one variable but the coefficients into K[x].
The problem with that is that while K is a field, we know that K[x] is not.
So that's a nice question.
But yes, you can prove that every ideal I = (p) where p is an irreducible element inside a Ring is maximal, then the result which involves the quotient R/M, that you can see on the left side of the blackboard, is true without making any assumptions on the qualities R must have.
So yes, that should estend onto the multi variable case
Tku