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Math Help - Rings exam problem

  1. #1
    Junior Member
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    Nov 2008
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    Rings exam problem

    Let \mathbb{K} be a field, consider the ring \mathbb{K}[x,y].

    1) Let p(x,y) \in \mathbb{K}[x,y], prove that there exist q(x,y) \in \mathbb{K}[x,y] and a(x), b(x) \in \mathbb{K}[x] unique such that p(x,y)=(x^3-y^2)q(x,y)+a(x)+b(x)y.

    2)Prove that the map \phi : \mathbb{K}[x,y]\longrightarrow \mathbb{K}[t] defined by \phi(p(x,y)) = p(t^2,t^3) is a ring morphism.

    3)Find the kernel of \phi (i.e: find generators of it)

    4) Find explicitly the image of \phi

    5)Prove that x^3-y^2 is irreducible in \mathbb{K}[x,y] . Deduce that the image of \phi is a domain.



    My views:
    1) No clue. Iīm thinking that it has to do with some division but I donīt know what...

    2) Ugly computation, I used the identity given in part 1) and I think I pulled it off (if thereīs a simpler way of doing it please let me know).

    3) It should be <x^3-y^2> I suppose... Am I right?

    4) p(t^2,t^3) = a(t^2)+t^3b(t^2) means something??

    5) Maybe I can use that \frac{\mathbb{K}[x,y]}{<x^3-y^2>} \cong Im\phi once I proved that x^3-y^2 is irreducible, but I donīt know how (Eisenstein doesnīt work, I think).


    Thanks, guys!!
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Paris
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    Hi

    1) Division, you said it! Use the fact that K[x][y]=K[x,y].
    -1 is invertible in K[x], so \forall p\in K[x][y],\ \exists !q,r\in K[x][y] such that:
    p(x,y)=(-y^2+x^3)q(x,y)+r(x,y)\ , and deg(r)<1, i.e. r(x,y)=b(x)y+a(x) for two unique a,b\in K[x]. (can be written bette, when I write the identity p(x,y) refers to a P(y) whose coefficients are in K[x] so it's equal to a unique polynomial in K[x,y] and there is no problem, same thing fo q and r)

    2) is quite easy! I don't think you really need 1)

    3) Here 1) can be useful. You've guessed it, but how to prove that? Take a p\in K[x,y], assume it belongs to ker\phi , and write it with the identity from 1). Your goal is to prove that a(x)+<br />
b(x)y=0

    4)Take a n\in\mathbb{N}. Is t^n in Im\phi ?

    5)Suppose -y^2+x^3\in K[x][y] is reducible, i.e. \exists\alpha ,\beta\in K[x]\ \text{s.t.}\ -y^2+x^3=-(uy+\alpha(x))(u^{-1}y+\beta(x)).
    Then u^{-1}\alpha(x)+u\beta(x)=0 and \alpha(x)\beta(x)=x^3 since x is irreducible in K[x] which is factorial we have \alpha(x)=vx^i and \beta(x)=v^{-1}x^{3-i} for an invertible v and a i\in\{0,1,2,3\}. This makes u^{-1}\alpha(x)+u\beta(x)=0 impossible, contradiction.
    Therefore x^3-y^2 is irreducible in K[x][y]=K[x,y], and you go on as you said.
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