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Problem:

Is $f(x) = x^5 + x^3 + 1$ irreducible in $\mathbb{F}_{32}$ and $\mathbb{F}_8$?

My thought:

$f(x)$ is irreducible in $\mathbb{F}_2$ and has degree $5$. So we can conclude that $\mathbb{F}_{32} \simeq \mathbb{F}_2[x]/f(x)$. Then apparently $f$ is not irreducible in $\mathbb{F}_{32}$.

But I don't know how to work on the case $\mathbb{F}_8$. I know that $\mathbb{F}_8 \simeq \mathbb{F}[x]/g(x)$ where $g(x)$ is some irreducible polynomial of degree $3$ in $\mathbb{F}_2 [x]$. For example, it can be $g(x) = x^3 + x + 1$. But how would that help me?

3x89g2
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2 Answers2

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Let $g(x)$ be an irreducible factor of $f(x)$ over $\mathbb F_8$, so $f(x)=g(x)h(x)$ for some $h(x)\in\mathbb F_8[x]$. From the inclusion of fields $\mathbb F_2\rightarrow\mathbb F_8$ we obtain an inclusion $\mathbb F_2[x]\rightarrow\mathbb F_8[x]$, giving a homomorphism $$ \phi:\mathbb F_2[x]\rightarrow\mathbb F_8[x]/g(x). $$ Now $\phi(f(x))=g(x)h(x)+g(x)\mathbb F_8[x]=0+g(x)\mathbb F_8[x]$, so $\phi$ induces a homomorphism $$ \bar\phi:\mathbb F_2[x]/f(x)\rightarrow\mathbb F_8[x]/g(x). $$ Since $f(x)$ is irreducible over $\mathbb F_2$ and $g(x)$ over $\mathbb F_8$, we have $$ \mathbb F_2[x]/f(x)\cong\mathbb F_{32},\hspace{10mm} \mathbb F_8[x]/g(x)\cong\mathbb F_{8^d} $$ where $d=\deg g(x)$. Therefore $\bar\phi$ gives a homomorphism $$ \mathbb F_{32}\rightarrow\mathbb F_{8^d}. $$ In particular $\mathbb F_{8^d}$ is a vector space over $\mathbb F_{32}$, so $$ 5=\log_2(32)\mid\log_2(8^d)=3d. $$ Hence $5|d$, so $d\geq5$. Thus $f(x)$ must be a scalar multiple of $g(x)$, and in particular $f(x)$ is irreducible over $\mathbb F_8$.

stewbasic
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  • I don't know exactly what you did but your answer looks cool to me, so can you explain a bit more about the machinery behind this solution? It will help me to learn. – Kushal Bhuyan Jul 11 '16 at 01:47
  • @KushalBhuyan which step did you want more explanation about? – stewbasic Jul 11 '16 at 01:53
  • That homomorphism thing and how the last two line implies the irreducibility? Or is there a general result of this application ? – Kushal Bhuyan Jul 11 '16 at 01:55
  • If $A, B$ are rings and $I$ an ideal of $A$ then a homomorphism $f:A\rightarrow B$ such that $f(I)=0$ induces a homomorphism $A/I\rightarrow B$. I applied this with $A=\mathbb F_2[x]$, $B=\mathbb F_8[x]/g(x)$ and $I=(f(x))$. For the last part, if $f(x)$ were reducible it would have an irreducible factor with degree $<5$, but I showed that any such factor has degree $\geq5$. – stewbasic Jul 11 '16 at 02:21
  • How did you come up with the second homomorphism? Why does it exist? I am comfortable with the first homomorphism. – 3x89g2 Jul 11 '16 at 22:43
  • @Misakov I added some more details, does that clarify? – stewbasic Jul 11 '16 at 23:24
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Here’s another approach, based on your good start. You’ve observed that $f$ is $\Bbb F_2$-irreducible, and that indeed $\Bbb F_{32}$ is its splitting field. In other words, $f(x)=\prod_i(x-\alpha_i)$, where the alpha’s are five different elements of $\Bbb F_{32}$. If you could take fewer than five of the factors above and multiply them together to get a polynomial in $\Bbb F_8[x]$, then its coefficients would be in $\Bbb F_{32}\cap\Bbb F_8=\Bbb F_2$, in other words a proper $\Bbb F_2$-divisor of $f$. Thus the only way you can multiply some of those factors together to get an $\Bbb F_8$-polynomial is to use all of them. So $f$ is $\Bbb F_8$-irreducible.

Lubin
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  • "Thus the only way ... is to use all of them". Is this simply because if that's not the case, then you would have $g(x) \in \mathbb{F}_2 [x]$ that has degree less than $f$ and divides $f$, which contradicts the fact that $f$ is irreducible in $\mathbb{F}_2 [x]$? – 3x89g2 Jul 11 '16 at 22:04
  • Right you are, @Misakov. – Lubin Jul 12 '16 at 01:41