TITLE: Is there an irreducible but solvable septic trinomial $x^7+ax^n+b = 0$?
QUESTION [19 upvotes]: The following irreducible trinomials are solvable:
$$x^5-5x^2-3 = 0$$
$$x^6+3x+3 = 0$$
$$x^8-5x-5=0$$
Their Galois groups are isomorphic to ${\rm D}_5$, ${\rm S}_3 \wr {\rm C}_2$ and
$({\rm S}_4 \times {\rm S}_4) \rtimes {\rm C}_2$, respectively.
Question: Is there an irreducible septic trinomial $x^7+ax^n+b=0$ with
solvable Galois group, where $n \in \{1, \dots, 6\}$, $a, b \in \mathbb{Z} \setminus \{0\}$? 
P.S. For $n=1$, I did a search only for those with 1 real root and, if I did it correctly, there are none with integer $|a,b|<50$.

REPLY [5 votes]: The good news is that in this book:
Generic Polynomials: Constructive Aspects of the Inverse Galois Problem
 By Christian U. Jensen, Arne Ledet, Noriko Yui (pp. 52 and up).
There is a complete criterion, as follows:
$x^7 + a x + b$ is solvable (we assume it's irreducible) if and only if the polynomial (which I am intentionally leaving in Mathematica input form, so you can type it in yourself).
P35[a_, b_, x_] := 
 x^35 + 40 a x^29 + 302 b x^28 - 1614 a^2 x^23 + 2706 a b x^22 + 
  3828 b^2 x^21 - 5072 a^3 x^17 + 2778 a^2 b x^16 - 
  18084 a b^2 x^15 + 36250 b^3 x^14 - 5147 a^4 x^11 - 
  1354 a^3 b x^10 - 21192 a^2 b^2 x^9 - 26326 a b^3 x^8 - 
  7309 b^4 x^7 - 1728 a^5 x^5 - 1728 a^4 b x^4 + 720 a^3 b^2 x^3 + 
  928 a^2 b^3 x^2 - 64 a b^4 x - 128 b^5
has factorization pattern [degrees of irreducible factors] one of (14, 21), (7, 7, 7, 21), (7, 7, 7, 14), (7, 7, 7, 7, 7)
The bad news is that I have run this for all pairs (a, b) where both coordinates are between -1000 and 1000, and there is not a single success.