irreducibility of discriminant

Question

This must be well-known to everyone but me, but here goes: take a general (monic) polynomial $p(x) = x^d + a_{d-1} x^{d-1} + \dotsc + a_0.$ The discriminant is a polynomial $D(a_0, \dotsc, a_{d-1}).$ Is this irreducible (over $\mathbb{C},$ or in general over algebraic closure of whatever the polynomial is defined over)?

Answer 1

The discriminant locus has the following geometric interpretation, given in the introductory chapter of [Gelfand, Kapranov, Zelevinsky: Discriminants, Resultants and Multidimensional Determinants].

Let $X \subset \mathbb{P}^d$ be the rational normal curve, $[u:v] \mapsto [u^d:u^{d-1}v:\cdots : v^d]$. We may view a linear form $l \in \Gamma(\mathcal{O}_{\mathbb{P}^d}(1))$ as filling in coefficients for a degree-$d$ polynomial, viz. restricting $l$ to $X$. This identifies the space of non-zero polynomials (modulo scalars) of degree $\leq d$ as the projective linear dual $P^{\vee}$ of $P := \mathbb{P}^d$. The roots of a polynomial are the (preimages on $\mathbb{P}^1 \hookrightarrow \mathbb{P}^d$ of) the intersection points of $X$ with the hyperplane $\{l = 0\}$. The discriminant locus is then the closure $X^{\vee} \subset P^{\vee}$ of the set of all the hyperplanes of $P$ that are tangent to $X$.

This gives the conceptual reason, since the following straightforward geometric argument (Prop. 1.3 in [GKZ]) proves in general that $X^{\vee} \subset P^{\vee}$ is irreducible whenever $X \subset P = \mathbb{P}^d$ is irreducible. Assuming WLOG that $X$ is regular, the incidence variety $W \subset P \times P^{\vee}$ of pairs $(x,H)$ with $x \in X$ and $H$ a hyperplane tangent to $X$ at $x$, is a projective bundle over the irreducible variety $X$, and hence irreducible. It follows that $X^{\vee}$, being the image $W \hookrightarrow P \times P^{\vee} \to P^{\vee}$ of an irreducible variety, is itself irreducible.

The geometric interpretation applies to more general discriminants, studied in [GKZ].

Answer 2

(N.B.: I have modified this answer to take into account the comments below about the case of characteristic $2$, when in fact, the discriminant is the square of an irreducible polynomial.)

The discriminant is defined by the property that, when $$ p(x) = x^d - s_1 x^{d-1} + \cdots + (-1)^ds_d = (x-t_1)(x-t_2)\cdots(x-t_d) $$ (i.e., when one substitutes the $i$th elementary symmetric function of the $t_k$ for $s_i$), the discriminant becomes $$ D(s_1,\ldots,s_d) = \prod_{i<j} (t_i-t_j)^2. $$ Suppose there were a factorization $D(s_1,\ldots,s_d) = D_1(s_1,\ldots,s_d)D_2(s_1,\ldots,s_d)$ where each $D_i$ had positive degree. Then by unique factorization, when one substitutes as above, one must be able to write, for $a = 1,2$, $$ D_a(s_1,\ldots,s_d) = c_a \prod_{(i,j)\in S_a} (t_i-t_j) $$ where $c_1$ and $c_2$ are (nonzero) constants and $S_1$ and $S_2$ are disjoint nonempty subsets of the set of pairs $(i,j)$ in $\{1,\ldots,d\}$ whose union is the entire set of distinct pairs in this set. Thus, for example, $D_1(s_1,\ldots,s_d)$ will vanish when $t_i=t_j$ (for $i\not=j)$ only if $(i,j)$ or $(j,i)$ belongs to $S_1$. Since one can't detect which of the $t_i$ are equal using only $s_1,\ldots, s_d$, it follows that $S_1$ must contain either $(i,j)$ or $(j,i)$ for each distinct pair. The same argument applied to $D_2$ shows that $S_2$ also must contain either $(i,j)$ or $(j,i)$ for each distinct pair. Thus, one must have, for $a = 1,2$, $$ D_a(s_1,\ldots,s_d) = c'_a \prod_{i<j} (t_i-t_j) $$ for some constants $c'_1$ and $c'_2$.

However, when the characteristic of the field is not $2$, one cannot have a polynomial $E(s_1,\ldots,s_d)$ such that, after substitution, one obtains $$ E(s_1,\ldots,s_d) = \prod_{i<j} (t_i-t_j) $$ since the left hand side cannot detect permutations in the $t_i$, whereas the right hand side will change sign when one makes an odd permutation of the $t_i$. Thus, $D$ is irreducible when the characteristic of the field is not $2$.

As Jarek Kuben pointed out in the comments below, when the characteristic of the field is $2$, the expression $$ F = \prod_{i<j} (t_i-t_j) = \prod_{i<j} (t_i+t_j) $$ is symmetric in the $t_i$, so $F$ can be written as a polynomial in the $s_i$. One then has $D = F^2$, so $D$ is a square. The above argument shows, however, that $F$ must be irreducible, since any factorization $D = D_1D_2$ has to have $D_1 = D_2 = F$ (up to constant multiples).

Answer 3

Some remarks:

1. Exception: In some cases it is actually reducible: Over a finite field of characteristic 2, the discriminant of $x^2+a_1x+a_0$ is $D(a_0,a_1)=a_1^2$.

2. (Almost) Irreducibility over finite fields: Standard $\zeta$-function manipulations show that over $\mathbb{F}_q[T]$ there are exactly $q^{d-1}$ monic non-squarefree polynomials (hint: the Dirichlet series of $\mu^2(f)=1_{f\text{ squarefree}}$ is $\frac{\zeta(u)}{\zeta(u^2)}$ where $\zeta(u) = \sum_{f \text{ monic}} u^{\deg f}$). In other words, the zero locus of $D(a_0,\cdots,a_{d-1})=0$ is of size $q^{d-1}$. On the other hand, Weil-Lang bounds express the size of the zero locus of $D=0$ over $\mathbb{F}_{q_0}$ as $q_0^{d-1} \cdot (1+O(\frac{1}{\sqrt{q_0}}))$ times the number of geometrically irreducible components of $D=0$ of dimension $d-1$. So using the information from the zeta functions for $q=q_0^k, k \to \infty$, we find that $D$ is a power of an irreducible polynomial.

3. (Almost) Irreducibility over $\overline{\mathbb{Q}}$: This reduces to irreducibility over number fields, which reduces to irreducibility over rings of integers (Gauss lemma) which follows from irreducibility over finite fields by working modulo some prime ideal.