# Irreducible polynomial over $\mathbb{Q}[x]$

Question:

Check if: $$f(x) = x^4 + 4x^3 + 6x^2 + 2x + 1$$ is reducible or irreducible over $$\mathbb{Q}[x]$$

Suppose that $$f(x)$$ is irreducible over $$Z_p[x]$$ for a prime $$p$$, therefore it's going to be irreducible over $$\mathbb{Q}[x]$$. Let $$p=2$$. Therefore: $$f_2(x) = f(x) \mod{2} = x^4 + 1 = a_2(x)\cdot b_2(x) \in \mathbb{Z}_2[x]$$ Since $$x^4+1$$ have roots in $$\mathbb{Z}_2[x]$$, there are two cases to check... First let's suppose that $$a_2(x),b_2(x)$$ have degree equal to two, therefore: \begin{align*} x^4+1&=(a_2x^2+a_1x+a_0)\cdot(b_2x^2+b_1x+b_0)=\\ &=(a_2b_2)x^4+(a_2b_1+a_1b_2)x^3+(a_2b_0+a_1b_1+a_0b_2)x^2+(a_1b_0+a_0b_1)x^+(a_0b_0) \end{align*} Since $$a_i,b_i \in \mathbb{Z}_2 \rightarrow a_i,b_i\in \{0,1\}$$, it follows that: $$a_0b_0=1 \longrightarrow a_0=b_0=1\\ a_1+b_1=0 \longrightarrow a_1=b_1=0\\ a_2b_2=1 \longrightarrow a_2=b_2=1$$ Hence: \begin{align*} a_2(x)&=x^2+1\\ b_2(x)&=x^2+1\\ a_2(x)\cdot b_2(x) &= (x^2+1)^2 = x^4 + 2x + 1 \overbrace{\longrightarrow}^{\in \mathbb{Z}_2} x^4 + 1 \end{align*}

Now let's check WLOG the case for $$a_2(x)$$ having degree equal to $$3$$ and $$b_2(x)$$ having degree equal to $$1$$:

\begin{align*} x^4+1 &= a_2(x)\cdot b_2(x) = (a_3x^3+a_2x^2+a_1x+a_0)\cdot(b_1x+b_0)\\ &= (b_1a_3)x^4 + (b_1a_2+b_0a_3)x^3 + (b_1a_1+b_0a_2)x^2 + (b_1a_0+b_0a_1)x + (b_0a_0) \end{align*}

By the same reasoning as before: \begin{align*} b_0a_0=1 \longrightarrow b_0=a_0=1\\ b_1a_3=1 \longrightarrow b_1=a_3=1\\ a_2+b_0 = 0 \longrightarrow a_2=b_0=0\\ a_1+a_2 = 0 \longrightarrow a_1=a_2=0\\ \end{align*}

Conclusion, it is reducible over $$\mathbb{Z}_2[x]$$ and therefore I can't conclude anything.

• Seems to me you have shown that $f(x)$ is reducible over $\Bbb Z_2[x]$, not whether or not it's reducible over $\Bbb Q[x]$. – Rolf Hoyer Dec 6 '18 at 3:34
• @RolfHoyer Yeah, I was checking what I've done... I just know that if a monic polynomial is reducible over $\mathbb{Q}[x]$, then it is going to be reducible over $\mathbb{Z}_2[x]$... But if that polynomial is irreducible, I can't assume the same... How you would approach that question? – Bruno Reis Dec 6 '18 at 3:37
• Why doesn't $x^4+1$ have roots mod $2$? – Michael Burr Dec 6 '18 at 3:38
• @MichaelBurr in mod 2, $x \in \{0,1\}$ and from that we have $f(0) = 1$ and $f(1)=2=0$... Yeah, you are correct... We have roots... – Bruno Reis Dec 6 '18 at 3:39
• @AnikBhowmick Shifting is an invertable operation, so irreduciblity must be preserved. In this case, factoring commutes with shifting. – Michael Burr Dec 6 '18 at 3:45

The expression for $$f(x)$$ looks pretty much like a binomial expansion: to be precise, $$f(x)=(x+1)^4-2x\ .$$ Hence $$f(x-1)=x^4-2x+2\ ;$$ this is irreducible by Eisenstein, therefore $$f(x)$$ is also irreducible.
Your polynomial is very close to $$(x+1)^4$$. Making the substitution, $$x \rightarrow y-1$$, we get $$y^4 - 2y + 2$$, which is irreducible by Eisenstein's criterion.