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Homework Help: Rings and fields

  1. Aug 1, 2010 #1
    1. The problem statement, all variables and given/known data

    is the set of all polynomials a ring,and a fieldd.Is is commutative and does it have unity

    2. Relevant equations



    3. The attempt at a solution

    now if we add or multiply any polynomials we get a polynomial. So it is a ring, but i am not sure what the multiplicative inverse is or whether every non zero element has an inverse to constitute a field.
     
  2. jcsd
  3. Aug 1, 2010 #2

    HallsofIvy

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    Multiplication is just ordinary multiplication of polynomials and the identity is the "constant" polynomial p(x)= 1 for all x. Does there exist a polynomial p(x) such that (x-1)(p(x))= 1? Think about the degree of p and what multiplication of polynomials does to degrees.
     
  4. Aug 1, 2010 #3
    i am thinking that we do not have multiplicative inverses because there is no polynomial that would give one, we would need to use a rational function, is this a correct assesmnent?
     
  5. Aug 1, 2010 #4

    HallsofIvy

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    Yes, but to give a complete answer, you need to say why "no polynomial would give one" (I presume you mean there is no polynomial, p(x), such that (x-1)p(x)= 1.)

    Again, think of the "degree" of polynomials. The degree of x- 1 is 1. What could the degree of (x-1)p(x) be? Could it be equal to 0, the degree of the constant polynomial, 1?
     
  6. Aug 1, 2010 #5
    ok thanks
     
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