Let $p(x)$ be the chromatic polynomial of a special graph. Performing a certain type of operation on the graph changes $p$ by shifting it and adding a constant, say to: $q(x)=p(x+a) + b$. 


I have noticed that this operation always results in $q$ having the same discriminant and splitting field as $p$. This would of course be obvious if we were just shifting $p$ by $a$, but it seems unusual given that we are also adding $b$.


I realise it is difficult to comment without knowing more details, but all I really want to know is whether or not this might be significant. The polynomials in question are irreducible cubics. How common is it that two cubics - which are not just shifted versions of each other - have the same discriminant? Does anybody have any idea what this might signify?


(edit: since posting this I have noticed that the chromatic polynomial in question is of the form $p(u,v,x)$, where $u$ and $v$ are given constants, and the discriminant of $p$ is a symmetric polynomial in $u$ and $v$. The graph operation switches $u$ and $v$...this is why the discriminant remains the same. So my question could be formulated as: what is the significance - if any - of a discriminant which is a symmetric polynomial in 2 variables?)



polynomials with the same discriminant
[Substantial edit: As I mentioned previously, cubic fields can have the same discriminant but not be isomorphic; but I've revised my answer to better address the author's question.]


There are a lot of polynomials that will generate the same cubic field. Here is the basic idea, due to Delone and Faddeev. Write down the cubic form $f(x, y) = ax^3 + bx^2 y + cxy^2 + dy^3$, where setting $y = 1$ gives a generating polynomial for the cubic field. Then there is an action of $GL\_2$ on such cubic fields; if $\gamma$ is a 2x2 matrix with entries $\alpha, \beta, \gamma, \delta$ then
$\gamma f(x, y) = f(\alpha x + \gamma y, \beta x + \delta y)$, optionally with a factor of $(det \ \gamma)^{-1}$.


Anyway, if two cubic forms are $GL\_2(\mathbb{Q})$-equivalent then the corresponding polynomials (if irreducible and nondegenerate) generate the same field. Moreover, if two cubic forms are $GL\_2(\mathbb{Z})$-equivalent then the corresponding polynomials generate the same order over $\mathbb{Z}$.


See [here](http://www.math.ucsd.edu/%7Ewgan/G2.ps), page 10 of Section 4, for a good reference.


[Previous answer:]


I think the underlying question is how often nonisomorphic cubic *fields* have the same discriminant. This happens, but it doesn't seem to be very common.


You might find it interesting to browse through [this database](http://hobbes.la.asu.edu/NFDB/) which lists cubic (and other) fields, their discriminants, and their minimal polynomials.


Theoretically, I believe it is known that this happens infinitely often, although I don't recall the reference. For upper bounds, it is known that the number of cubic fields of discriminant n is $O(n^{1/3 + \epsilon})$, due to Ellenberg and Venkatesh; the "correct" bounds are believed to be much sharper.


