  Conditional on the extended Riemann hypothesis, we show that with high
probability, the characteristic polynomial of a random symmetric $\{\pm
1\}$-matrix is irreducible. This addresses a question raised by Eberhard in
recent work. The main innovation in our work is establishing sharp estimates
regarding the rank distribution of symmetric random $\{\pm 1\}$-matrices over
$\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$. Previously, such
estimates were available only for $p = o(n^{1/8})$. At the heart of our proof
is a way to combine multiple inverse Littlewood--Offord-type results to control
the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$
with anticoncentration at least $1/p + \Omega(1/p^2)$. Previously, inverse
Littlewood--Offord-type results only allowed control over vectors with
anticoncentration at least $C/p$ for some large constant $C > 1$.
