Is $x^{2k+1} - 7x^2 + 1$ irreducible? 
Question. Is the polynomial $x^{2k+1} - 7x^2 + 1$ irreducible over $\mathbb{Q}$ for every positive integer $k$?

It is irreducible for all positive integers $k \leq 800$.
 A: Here is a proof, based on a trick that can be used to prove that
$x^n + x + 1$ is irreducible when $n \not\equiv 2 \bmod 3$.
We work with Laurent polynomials in $R = \mathbb Z[x,x^{-1}]$; note that
$R$ has unit group $R^\times = \pm x^{\mathbb Z}$.
We observe that for $f \in R$, the sum of the squares of the coefficients
is given by
$$\|f\| = \int_0^1 |f(e^{2 \pi i t})|^2\,dt
        = \int_0^1 f(e^{2 \pi i t}) f(e^{-2 \pi i t})\,dt
        = \int_0^1 f(x) f(x^{-1})\big|_{x = e^{2 \pi i t}}\,dt .$$
Now assume that $f(x) = g(x) h(x)$. Then, since
$f(x) f(x^{-1}) = \bigl(g(x)h(x^{-1})\bigr)\bigl(g(x^{-1})h(x)\bigr)$,
$G(x) := g(x) h(x^{-1})$ satisfies $\|G\| = \|f\|$
and $G(x) G(x^{-1}) = f(x) f(x^{-1})$.
Now we consider $f(x) = x^n - 7 x^2 + 1$; then $\|f\| = 51$.
If $f = g h$ as above, then write
$G(x) = \pm x^m(1 + a_1 x + a_2 x^2 + \ldots)$
and $G(x^{-1}) = \pm x^l(1 + b_1 x + b_2 x^2 + \ldots)$.
The relation $G(x) G(x^{-1}) = f(x) f(x^{-1})$ translates into
(equality of signs and)
$$(1 + a_1 x + \ldots)(1 + b_1 x + \ldots) = 1 - 7 x^2 + O(x^{n-2}).$$
Assuming that $n > 40$ and considering terms up to $x^{20}$, one can
check (see below) that the only solution such that
$a_1^2 + a_2^2 + \ldots + a_{20}^2 + b_1^2 + b_2^2 + \ldots + b_{20}^2\le 49$
is, up to the substitution $x \leftarrow x^{-1}$, given by
$1 + a_1 x + \ldots = 1 - 7x^2 + O(x^{21})$,
$1 + b_1 x + \ldots = 1 + O(x^{21})$. Since the $-7$ (together with the
leading and trailing 1) exhausts our allowance
for the sum of squares of the coefficients, all other coefficients
must be zero, and we obtain that $G(x) = \pm x^a f(x)$
or $G(x) = \pm x^a f(x^{-1})$. Modulo interchanging $g$ and $h$,
we can assume that $g(x) h(x^{-1}) = \pm x^a f(x)$,
so $h(x^{-1}) = \pm x^a f(x)/g(x) = \pm x^a h(x)$,
and $x^{\deg h} h(x^{-1})$ divides $f(x)$.
This implies that $h(x)$ divides $x^n f(x^{-1})$, so $h(x)$ must divide
$$f(x) - x^n f(x^{-1}) = 7 x^2 (x^{n-4} - 1).$$
So $h$ also divides
$$f(x) - x^4 (x^{n-4} - 1) = x^4 - 7 x^2 + 1 = (x^2-3x+1)(x^2+3x+1).$$
Since $h$ also divides $f$, it must divide the difference $x^n - x^4$,
which for $n \neq 4$ it clearly doesn't, since the quartic has no
roots of absolute value 0 or 1; contradiction.
The argument shows that $x^n - 7 x^2 + 1$ is irreducible for $n > 40$;
for smaller $n$, we can ask the Computer Algebra System we trust.
This gives:
Theorem. $x^n - 7 x^2 + 1$ is irreducible over $\mathbb Q$
for all positive integers $n$ except $n=4$.
ADDED 2017-01-08:
After re-checking the computations, I realized that there
was a small mistake that ruled out some partial solutions prematurely.
It looks like one needs to consider terms up to $x^{20}$.
Here is a file with MAGMA code that verifies that
there are no other solutions.
A: Not an answer, but an observation, which, if proved, will probably give an answer.
For all $k$ that I have computed, your polynomial has two roots smaller than $1$ in absolute value (for large $k$ these converge to $\pm\frac{1}{\sqrt{7}}.$), and the rest of the roots are larger than $1$ in absolute value. Let's assume this is true. Since the constant term of a factor has to have absolute value $1$ each factor should have one of these roots, so there are at most $2$ irreducible factors.
A: EDIT: I'm afraid that this answer is wrong and I hope you have some way of "unaccepting" it. The reason is that when taking conjugates of $\frac{1}{\sqrt{7}-\beta_1}$ we're allowed to freely change the sign of $\sqrt{7}$ (unless $\sqrt{7}$ is contained in the field generated by $\beta_1$, which seems rather unlikely). And $\frac{1}{-\sqrt{7}-\beta_1}$ is greater than $1$ in absolute value. I found this mistake only when I noticed that my "proof" gives that $x^4-7x^2+1=(x^2+3x+1)(x^2-3x+1)$ is irreducible. It should work with an integer square in place of $7$. Maybe someone can make it work also for $7$? I apologize for raising false hopes!
The question is equivalent (by replacing $x$ with $\frac{1}{x}$) to asking about the irreducibility of $A(x) = x^{2k+1}-7x^{2k-1}+1 = x^{2k-1}(x^2-7)+1$ over $\mathbb{Q}$. An application of Rouché's theorem shows that $A(x)$ has exactly $2k-1$ zeroes (counted with multiplicity) of absolute value less than $1$. There remain two real zeroes $\beta_1 < -1$ and $\beta_2 > 1$. One can deduce from the equation they satisfy that $|\beta_i - (-1)^i\sqrt{7}| \leq \frac{1}{\sqrt{7}+1}$ ($i=1,2$).
Now suppose that $A$ factorizes as $A = BC$ with monic integer polynomials $B, C$. Wlog $B(\beta_1) = 0$. If $B(\beta_2) \neq 0$, then
$$\beta_1^{2k-1}(\beta_1+\sqrt{7}) = \frac{1}{\sqrt{7}-\beta_1}$$
is an algebraic integer, all conjugates of which are less than $1$ in absolute value (the right hand side is less than $\frac{1}{2\sqrt{7}-\frac{1}{\sqrt{7}+1}}<1$ and all its conjugates are of absolute value at most $\frac{1}{\sqrt{7}-1}<1$).
So it would have to be $0$, which is absurd. So $B(\beta_2)=0$. But now all roots of $C$ are less than $1$ in absolute value, while its constant term is $1$. It follows that $C=1$ is constant and so $A$ is irreducible.
