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I need to show that if I have a reducible polynomial $f\in \mathbb{F}_{3}[X] $ of degree $5$ with no roots. Then there exists a monic irreducible polynomial of degree 2 dividing f.


My attempt:

$\mathbb{F}_{3}[X]$ is a unique factorization domain, so f can be factored uniquely by irreducible elements.
If f is factored by two polynomials $f=q_1 q_2$, then it needs to satisfy $\deg(f)=\deg(q_1)+\deg(q_2)=5$.
So if $p_n$ is an irreducible polynomial of degree n, then f can be factored in two ways: $f=p_1p_4$ or $f=p_2p_3$

(But do we don't know anything about the multiplicity of the roots? So couldn't it in theory be factored as a multiple of the same degree polynomial? For example 1 one degree and 2 second degree polynomials?)

I'm somehow supposed to conclude that the only way it can be factored is by $f=p_2p_3$ and thus there must be a second-degree polynomial dividing f (since it is a factor of a second degree). But how do I draw this conclusion?

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If the polynomial is reducible, it can be factored as $f=pq$ with $0\leq \deg(p),\deg(q)\lt \deg(f)$. In this case, $\deg(f)=5$. Moreover, you know that $f$ does not have any degree $1$ factors, since it has no roots and roots correspond to degree $1$ factors.

Now, the factorization of $f$ into irreducibles will correspond to a partition of $5$, given by the degrees of the irreducibles. The partitions of $5$ are: $$\begin{align*} 5 &= 5\\ 5 &= 4+1\\ 5 &= 3+2\\ 5 &= 3+1+1\\ 5 &= 2+2+1\\ 5 &= 2+1+1+1+1\\ 5 &= 1+1+1+1+1. \end{align*}$$

Of these, only two of them correspond to factorizations with no factor of degree $1$, and one of those two corresponds to the case where $f$ is irreducible. That leaves only one possibility.

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  • $\begingroup$ Thanks, that makes sense. I think I forgot that f doesn't have any roots along the way. $\endgroup$ – sjm23 Nov 29 '20 at 0:16
  • $\begingroup$ Just to be sure. What are the reason that a 1st degree factor corresponds to a root? If I think on regular polynomials, then it makes sense. Because from the fundamental theorem of algebra, a polynomial can be factored into it's roots. and the factors will be of the form $(x-\alpha)$ which is first degree polynomial. But is there another explanation? Especially since in algebra we are dealing with all kinds of polynomials (Or should I say all kind of rings). $\endgroup$ – sjm23 Nov 29 '20 at 0:30
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    $\begingroup$ @sjm23: What is a “regular polynomial”, and how does it differ from a polynomial? You have polynomials over a field. The Factor Theorem holds, which tells you that $f(a)=0$ if and only if $x-a$ divides $f(x)$. And if $bx-c$ is a factor of $f$, $b\neq 0$, then $f(x) = (bx-c)q(x)$ for some $q(x)$, and then what happens when you evaluate at $\frac{c}{b}$? $\endgroup$ – Arturo Magidin Nov 29 '20 at 0:32

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