# Difficulties with resultants and Tschirnhaus transformations

I'm experimenting with some polynomial equations, and as a test I'm working through the example given here:

http://www.oocities.org/titus_piezas/...

Basically, in order to eliminate the first two terms of the quintic $x^5-x^4-x^3-x^2-x-1$ we need to find a transformation $y=x^2+ax+b$ (such a polynomial transformation is called a *Tschirnhaus transformation*, after its first discoverer), for which $a$ and $b$ have the effect of producing a quintic equation in $y$ but without $y^4$ or $y^3$ terms. This is as far as I've got so far:

```
R.<a,b> = QQ[]
S.<y> = R[]
T.<x> = S[]
p = x^5-x^4-x^3-x^2-x-1
res = p.resultant(y-x^2-a*x-b)
rc = res.coefficients()
solve([rc[-2],rc[-3]],[a,b])
TypeError: a is not a valid variable.
```

Drat! So I have two questions: why is `a`

not a valid variable, and why can I not isolate individual coefficients of `res`

? I can obtain all the coefficients of `res`

, but I can't isolate just one, with something like `res.coeff(y,3)`

.

If I declare variables, then solve also works