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 RUclips 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 RUclips to see what would come up...and here you are. I'm looking forward to watching more of your stuff.
5:00 is the cleanest edit
You can try to prove it yourself and skip ahead to 8:47 to watch the applications.
Brother you are doing great work..Love from India 🇮🇳🇮🇳🇮🇳
Me too ,from India. Sir, greetings from India
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