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This is my first exercise on polynomal, can u explain me, step by step how can I resolve it? I'm good with theory about $Z_n$ and I know something about polynomials, but I haven't clear view and I really don't know how to proceed. For example, on the 3rd question, I know how to determine $f(1)$ etc, and I also know when a polynomial is irreducibile but I don't know how to answer.

  1. What's the maximum number of possible roots that (in $\mathbb{Z}_{13}$) a polynomial with degree of ten and coefficient in $\mathbb{Z}_{13}[x]$ can have
  2. Determine (if possible) two distinct polynomials $u$ and $v$ in $\mathbb{Z}_{31}$, both of them with degree of twenty such that the set $\{a:\in\mathbb{Z}_{31}[x]: u(a) = v(a)\}$ have 25 elements.
  3. The polynomial $f=x^5+2x^4+10x+9\in\mathbb{Z}_{11}[x]$. Determine $f(1)$, $f(-1)$, $f(2)$, $f(-2)$, and says if $f$ has an irriducible factor with degree 3 in $\mathbb{Z}_{11}[x]$

Thank you.

Best regards

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    I'm good with theory about $Z_n$ and I know something about polynomials, but I haven't clear view and I really don't know how to proceed. For example, on the 3rd question, I know how to determine $f(1)$ etc, and I also know when a polynomial is irreducibile but I don't know how to answer.2019-02-13

1 Answers 1

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Hint $\ $ All are immediate consequences of the fact that a nonzero polynomial over a field (or domain) has no more roots than its degree. See here for a proof. In $(2)$ consider the polynomial $u - v$ and in $(3)$ consider $f/g,$ where $g$ is an irreducible cubic factor of $f$.

The point of the exercises is to help you recognize how this result applies in slightly perturbed contexts where the polynomials are differences or quotients.

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    @Mariano Yes indeed.2012-03-29