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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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3  
Think about accepting answers to your previous questions. –  xen Mar 27 '12 at 21:27
    
Is this a homework? If so, then please add the (homework) tag. Also, would please edit your post, and add info about what did you try? –  user2468 Mar 27 '12 at 21:33
    
Are you sure this is your first exercise ever on polynomials? It would help if you could give some clues about what you know about polynomials, and about $\mathbb{Z}_{13}, \mathbb{Z}_{31}, \mathbb{Z}_{11}$ and how you may have tried to deploy that knowledge in attempting to answer the question. Hint: the answer in each case is probably less complex than you imagine - it is an exercise in seeing how something simple that you know can be applied in apparently complex circumstances. You will learn hugely more by trying yourself than by getting an answer here. –  Mark Bennet Mar 27 '12 at 21:39
    
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. –  Mariano Mar 27 '12 at 21:58

1 Answer 1

up vote 3 down vote accepted

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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So the answer for 1) is 10. Can u improve your hint about 2) and 3)? What does it mean consider $u-v$, and how can I dind the cubic factor? What's the iter? –  Mariano Mar 28 '12 at 21:14
    
@Mariano $u(a)=v(a)\iff (u-v)(a) = 0.\:$ What is the degree of $u-v$? For $(3)$, note that a root of $f$ cannot be a root of $g$ since it is irreducible, so it must be a root of $f/g$, which has degree $\ldots$ so at most $\ldots$ roots. –  Bill Dubuque Mar 28 '12 at 21:25
    
degree of $\vartheta(u-v)\le max(u,v)$. Am I right? –  Mariano Mar 28 '12 at 21:33
    
@Mariano $\max(\deg\:u,\deg\:v) \le 20 < 25\ $ so $\max(\deg(u-v)) < \ldots$ –  Bill Dubuque Mar 28 '12 at 21:37
1  
@Mariano Yes indeed. –  Bill Dubuque Mar 29 '12 at 13:29

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