# Irreducibility of $x^4 + x^3 + 1$ over finite field $\mathbb{F}_{2^{a}}$, $1 \leq a \leq 6$

I have to discuss the irreducibility of $$P(X) = X^4 + X^3 + 1$$ over finite field $$\mathbb{F}_{2^{a}}$$, $$1 \leq a \leq 6$$.

So, for $$a = 1$$, we have that $$P$$ is irreducible since is has no roots in $$\mathbb{F}_2$$ and not the square of the only irreducible quadratic polynomial over $$\mathbb{F}_2$$ which is $$X^2 + X + 1$$

And if $$P$$ were reducible over $$\mathbb{F}_{2^b}$$, where $$b$$ is odd $$< 6$$, then each of its roots will generate $$\mathbb{F}_{2^4}$$, but this is a contradiction as $$\mathbb{F}_{p^k}$$ is a subfield of $$\mathbb{F}_{p^k}$$ if and only if $$k | n$$; this is clearly impossible, since $$b$$ is odd.

Therefore, for $$a = 3,5$$ i.e. over $$\mathbb{F}_{8}$$ and $$\mathbb{F}_{32}\ P$$ is irreducible; and it follows from the above discussion that $$P$$ is reducible over $$\mathbb{F}_{16}$$, since a root of this polynomial generates the field.

Now, if we let $$\mathbb{F}_4 = \{0,\ 1,\ a,\ a+1\ \vert\ a^2 + a + 1=0\}$$ then we see that $$P(X) = (X^2 + aX +a)(X^2 + (a+1)X + (a+1))$$, hence $$P$$ is reducible in $$\mathbb{F}_4$$

Which leaves irreducibility of $$P$$ over $$\mathbb{F}_{64}$$. It is here that I'm drawing a blank. Any help is appreciated!

• "...the only quadratic irreducible polynomial..." ...and then you wrote down a cubic ... – DonAntonio Nov 19 '18 at 0:53
• Ah, a silly mistake on my part. Corrected – Naweed G. Seldon Nov 19 '18 at 0:55

$$P$$ is reducible over $$\mathbb F_{64}$$, because reducible over the subfield $$\mathbb F_4\subset \mathbb F_{64}$$ ($$2\mid6$$).
• Does it also follow that $P$ has no roots over $\mathbb{F}_{64}$ – Naweed G. Seldon Nov 19 '18 at 2:05
• Such a root would generate $\mathbb F_{16}$, right? But $4\not\mid6$. – Chris Custer Nov 19 '18 at 2:28
$$\Bbb{F}_{64}\not \supset\Bbb{F}_{16}$$ which you know is the splitting field of $$P$$, but what about $$\Bbb{F}_{4096}$$? It's Galois over $$\Bbb{F}_{64}$$ and contains $$\Bbb{F}_{16}$$, the splitting field of $$P$$. What's it's degree over $$\Bbb{F}_{64}$$? What does that tell you about $$P$$ over $$\Bbb{F}_{64}$$?