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Math Help - Polynomial ring

  1. #1
    No one in Particular VonNemo19's Avatar
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    Polynomial ring

    A polynomial with integer coeffiecients is an expression of the form f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 where a_i\in\mathbb{Z}. If a_n\neq0, f(x) is said to be of degree n. Denoting the set of all such polynomials as \mathbb{Z}[x], show that \mathbb{Z}[x] is a ring with respect to a natural definition of addition and multiplication of polynomials.

    So, Let f_1(x),f_2(x)\in\mathbb{Z}[x], where f_1(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 and

    f_2(x)=b_kx^k+b_{k-1}x^{k-1}+...+b_1x+b_0. Now we have

    f_1+f_2=(a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0)+(b_kx^k+b_{k-1}x^{k-1}+...+b_1x+b_0)=.

    Clearly, anwhere n=k, the coefficient will be a_n+b_k, which is an integer. All of the rest of the terms will remain unchanged and so will necessarily have integer coefficients, therefore f_1+f_2\in\mathbb{Z}[x] and we conclude that the sum of two functions in \mathbb{Z}[x] is also in \mathbb{Z}[x].

    OK. I'm trying here but I'm stuck. Help?
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  2. #2
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    A polynomial of degree k that normally has k + 1 terms can be considered as having n + 1 terms for any n >= k; just make the coefficients b_n,\dots,b_{k+1} to be zero. The definition does not say that coefficients have to be non-zero. We are not talking about a degree here. Thus, when we need to add two polynomials, we can assume that they have the same number of terms.

    Do you doubt that the sum of two polynomials with integer coefficients is another such polynomial? For example, what about (5x^3+6x^2-3x-8)+(-3x^2+x+3)?
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  3. #3
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by emakarov View Post
    A polynomial of degree k that normally has k + 1 terms can be considered as having n + 1 terms for any n >= k; just make the coefficients b_n,\dots,b_{k+1} to be zero. The definition does not say that coefficients have to be non-zero. We are not talking about a degree here. Thus, when we need to add two polynomials, we can assume that they have the same number of terms.

    Do you doubt that the sum of two polynomials with integer coefficients is another such polynomial? For example, what about (5x^3+6x^2-3x-8)+(-3x^2+x+3)?
    No, I don't doubt that the sum of polynomials with integer coefficients will result an anything but another polynomial with said coeffiecients. I'm new to the subject and therefore I am having trouble with answering the questions. How would you show that \mathbb{Z}[x] is a ring wrt multiplication and addition? I would appreciate your help. Thank you.
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  4. #4
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    One has to check, first, that \mathbb{Z}[x] is closed under the operations, i.e., addition and multiplication do not return results that are outside of \mathbb{Z}[x], and, second, all the ring axioms.

    The definition of addition and multiplication is given here, from where it is pretty clear that \mathbb{Z}[x] is closed under these operations. Axioms about addition are pretty simple. Associativity of multiplication and distributivity can be proved directly from the definitions above by working with big sums, but there may be a more elegant way. I'll think about this more.
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