theory about polynomial, how can I resolve this exercise?

Question

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.

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

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.

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

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. — BAD_SEED, Mar 27 '12 at 21:58

score 3 · Accepted Answer · edited Apr 13 '17 at 12:20

3

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.

edited Apr 13 '17 at 12:20

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answered Mar 27 '12 at 21:59

Bill Dubuque

272,048

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? – BAD_SEED 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? – BAD_SEED 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
24? $u = a_0 + a_1x+ ... +a_{24}x^{24}$? – BAD_SEED Mar 28 '12 at 21:41
@Mariano Do you think that $u - v$ can have degree higher than $u$ or $v$? Note: please omit the second max in my prior comment. – Bill Dubuque Mar 28 '12 at 22:00
I don't know what to think. For sure it cannot be higher than u or v itself (I was wrong cause I've ignored 20 and just read $\leq 25$) – BAD_SEED Mar 28 '12 at 22:24
@Mariano Right. Now, since $u-v$ has degree at most $20$, can it have $25$ distinct roots? – Bill Dubuque Mar 28 '12 at 22:28
absolutly not, at most 20 roots. – BAD_SEED Mar 28 '12 at 22:33
So the answer for 2) is: it's impossible find polynomial of 25 elements with degree of 20? – BAD_SEED Mar 29 '12 at 07:18
1

@Mariano Yes indeed. – Bill Dubuque Mar 29 '12 at 13:29

theory about polynomial, how can I resolve this exercise?

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