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Polynomial ring

  1. Jan 30, 2009 #1
    "Let f is a polynomial from k[x,y], where k is a field. Suppose that x appears in f with positive degree. We view f as an element of k(y)[x], that is polynomial in x whose coefficients are rational functions of y."

    I think I am missing something...why do we need rational functions here? can't we represent any polynomial from k[x,y] as an element of k[y][x] i.e. polynomial in x whose coefficients are polynomials of y?
     
  2. jcsd
  3. Jan 30, 2009 #2

    jambaugh

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    I'm guessing here given I don't know from where you are getting this but...

    Recall that a polynomial is also a rational function just as an integer is also a rational number.

    I think the idea is to keep the format of k(y)[x] as a ring of polynomials (in x) over a field. So the y-polynomial coefficients are treated as elements of the larger ring of rational functions of y.
     
  4. Jan 30, 2009 #3
    Yes, k[X,Y]=k[X][Y]=k[Y][X].

    k[X,Y] is NOT equal to k(Y)[X].
     
  5. Feb 1, 2009 #4
    yes, later on it says that "if f (polynomial from k[x,y]) is irreducible then it remains irreducible as an element of k(y)[x]"

    can someone explain to me why it is so?

    ...I guess, if we assume f = gh where g,h are from k(y)[x] then we can multiply both sides by the product of common denominators of coefficients of g and h, So that we get f c(y) = g_1 h_1, where c(y), g_1, and h_1 are polynomials from k[x,y].

    Now, how does this imply that f is not irreducible??
     
  6. Feb 1, 2009 #5
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