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A few questions about a ring of polynomials over a field K

  1. Oct 29, 2015 #1
    1. The problem statement, all variables and given/known data

    Consider the ring of polynomails in two variables over a field K:

    [itex]R=K[x,y] [/itex]

    a)Show the elements x and y are relatively prime

    b) Show that it is not possible to write [itex]1=p(x,y)x+q(x,y)y [/itex]with [itex]p,q \in R [/itex]

    c) Show R is not a principle ideal domain

    2. Relevant equations
    None

    3. The attempt at a solution

    I'd like to give an attempt, but I have no idea on how to start.

    I know that two elements a,b are relatively prime if the only integer that divides them is 1.

    Any help would be greatly appreciated.
     
    Last edited: Oct 29, 2015
  2. jcsd
  3. Oct 29, 2015 #2

    andrewkirk

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    For (a) you need to show that if ##x=p_x d## and ##y=p_y d## where ##p_x,p_y,d## are polynomials over K then ##d=1##.

    Write each of the three polynomials as a finite sum of allowable terms, eg ##p_x=\sum_{i=0}^{m_x} \sum_{j=0}^{n_x} a_{ij}x^iy^j##.

    Then multiply out and equate coefficients.

    By the way, what is the meaning of ##p(x,y)## and ##q(x,y)## in (b)?
     
  4. Oct 29, 2015 #3
    Hmm, ok. I see what I can do. Thanks!

    p(x,y) and q(x,y) are polynomials in terms of x and y
     
  5. Oct 29, 2015 #4

    andrewkirk

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    Then they need to put more constraints on them for (b) to be true, because p(x,y)=1, q(x,y)=0 are two such polynomials that satisfy the equation. They are zero-order, but still polynomials.
     
  6. Oct 29, 2015 #5
    ohh sorry, thats my bad, let me edit it...
     
  7. Oct 29, 2015 #6
    Which terms do I equate? When the polynomials are expanded it looks pretty bad...
     
  8. Oct 30, 2015 #7

    andrewkirk

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    First try to prove that the ##order(pq)=order(p)+order(q)##, where ##order(f)## for polynomial ##f## is the highest sum of exponents. This is very easy for polynomials in a single variable. It's slightly harder for polynomials in two variables, but still very doable. Focus on the terms with the highest order in each of the two polynomials, but note that there may be more than one such term in each of them (eg ##x^2+xy+y^2##).

    Once you have that, you know that if ##x=p_x d## the order of one of the factors must be 1 and that of the other must be zero. A polynomial of order zero is a constant, which is a unit (divisor of 1) in the ring of polynomials. That's all you need. Above, where I said in post #2 that you need to show that ##d=1##, I should have said you need to show that ##d## is a unit.
     
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