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Given $a,b \in \mathbb Q$ and $ab \neq 0.\,$ Let us consider trinomials $x^n+ax^k+b,\,$ with integers $n > k$, $n > 8$, and $gcd(n,k) = 1$.

Questions:

  1. Do you know examples of reducible and solvable other than the following ones? $$x^9+19x+20 = \\ \left(x+1\right) \left(x^4-2x^3+2x^2-4x+5\right) \left(x^4+x^3+x^2+3x+4\right)$$ $$x^9+80919x+495720 = \\ \left(x^3+6x^2+27x+72\right)\left(x^6-6x^5+9x^4+36x^3-27x^2-1458x+6885\right)$$ $$x^9+3x^4-4 = \\ \left(x-1\right) \left(x^8+x^7+x^6+x^5+x^4+4x^3+4x^2+4x+4\right)$$ $$x^{10}+297000000x-1846800000 = \\ \left(x^4-60x^2-300x+5400\right)\left(x^6+60x^4+300x^3-1800x^2+36000x-342000\right)$$
  2. Are there any irreducible and solvable examples?

Notes:

  1. If $x^n+ax^k+b$ is solvable then $1+ax^{n-k}+bx^n$ is too. For example $x^9+19x^8+25600000000$ can be obtained from $x^9+19x+20$. So we can focus only on values $ k < \frac{n}{2} $.

  2. Similar questions for $n=8$ and $n=7$ were already asked:

  3. The latter link contains the answer that there exist only finite number of solvable irreducible trinomials for $n=7$ and $n>8$. But there is no explicit example shown.

  4. Details of how I found the mentioned nonic and decic trinomials are available on: https://sites.google.com/site/klajok/polynomials/non-irreducible .

  5. I have checked that for $8 < n < 17$ and integer $|a|,|b| \leq 100000$ there is no solvable irreducible trinomials.

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    +1. The $x^9+3x^4-4$ is beautiful. Its octic factors over $\sqrt{-3}$. – 2017-01-16
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    The closest irreducible but solvable nonic I found is the quadrinomial, $$6^4\, x^9 - 30x^2 - 10x - 5 =0$$ which results from eliminating $u$ between, $$ 6x^3 + 6u x^2 - 2(3u^2 + 3u - 5)x + u = 0\\6u^3 - 10u + 5 = 0.$$ – 2017-01-16

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