How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$?

Question

Please, check my solution:

$f(x)$ is a polynom of degree $n=3 (\delta f=3) \Rightarrow f(x)=a_0 + a_1 x +a_2 x^2 + a_3 x^3$. We have three choices for each coeficient, so we can build $3^4 = 81$ polynoms over $\mathbb Z_{3}$. The reductible polynoms are in the shape $p(x)=(x- \alpha)(b_0 + b_1 x + b_2 x^2)$, so we can build $3.(2^3)=24$ reductible polynoms over $\mathbb Z_{3}$. Conclusion: there are 57 irreductible polynoms. Thank you!

Note that if $p(x)$ is irreducible so is $2p(x)$ (and $4p(x)\equiv p(x)$) so you should be expecting to find an even number. — Mark Bennet, Sep 08 '14 at 20:15
Do you count $x(x^2+1)$ as reducible? - you seem to be counting as if $b_i\neq 0$ — Mark Bennet, Sep 08 '14 at 20:21

score 1 · Answer 1 · answered Sep 08 '14 at 20:45

First I count the number of irreducible monic degree-2 polynomials over $\def\zt{\Bbb Z_3}\zt$. It suffices to count the monic polynomials and then multiply by 2. A reducible monic polynomial of degree 2 has the form $(x-a)(x-b)$ for $\{a,b\}\subset\{\bar0,\bar1,\bar2\}$. There are 6 possible choices for $\{a,b\}$. There are 9 monic polynomials ($x^2+px+q$ for $\langle p,q\rangle\in\{\bar0,\bar1,\bar2\}^2$), so 3 are irreducible.

Now I count the number of irreducible degree-3 polynomials over $\zt$. There are $27$ monic polynomials. A reducible monic polynomial factors as either $(x-a)P_2(x)$ where $P_2$ is irreducible or as $(x-a)(x-b)(x-c)$. There are $9$ of the former ($3$ choices of $a$ times 3 choices of $P_2$) and $10$ of the latter, totaling $19$. This leaves $27-19=8$ monic irreducible polynomials, so $16$ irreducible polynomials in all.

score 0 · Accepted Answer · answered Sep 08 '14 at 20:33

0

I assume you mean degree 3, not degree at most 3. We can assume they are monic. If you want the non-monic ones just multiply all these by 2. So I get these irreducible monic polynomials of degree 3 over $F_3$:

$x^3+2x+1$

$x^3+2x+2$

$x^3+x^2+2$

$x^3+x^2+x+2$

$x^3+x^2+2x+1$

$x^3+2x^2+1$

$x^3+2x^2+x+1$

$x^3+2x^2+2x+2$

Being lazy, I just asked Mathematica to print them out, but I seem to have less than you, so maybe I asked in a silly way.

answered Sep 08 '14 at 20:33

almagest

18,380

This does not address the question. – user133281 Sep 08 '14 at 20:38
In what way does it not? – almagest Sep 08 '14 at 20:40
No. He asked us to check his solution. The comments have already pointed out that (indirectly) that it is totally wrong. Now you have two solutions showing that the correct number is 8/16 depending on whether you count $x^3+2x+2$ and $2x^3+x+1$ as the same. – almagest Sep 08 '14 at 20:49

How many irreductible polynoms of degree $n=3$ are there over $\mathbb Z_{3}=\{\overline{0}, \overline{1}, \overline{2}\}$?

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