Concrete example of non-abelian class field theory - why Langlands program *is* a non-abelian class field theory?

Actually the exact cubic polynomial $f(x) = x^3 - x - 1$ you named is the subject of an old MO answer of Matthew Emerton's. Its splitting behavior is described by a Hecke eigenform of weight $1$ and level $23$ (the absolute value of the discriminant of $x^3 - x - 1$) which has a product formula

$$A(q) = q \prod_{n=1}^{\infty} (1 - q^n)(1 - q^{23n}).$$

The coefficient $a_p$ of $q^p$, for $p$ a prime $\neq 23$, is the trace of the Frobenius element at $p$ in the Galois group $S_3$ acting on the unique $2$-dimensional irreducible representation of $S_3$ (which corresponds to the Galois representation corresponding to the modular form above), which means

• $f(x)$ splits into linear factors $\bmod p$ iff the Frobenius element is the identity iff $a_p = 2$,
• $f(x)$ splits into a linear and a quadratic factor $\bmod p$ iff the Frobenius element is a $2$-cycle iff $a_p = 0$, and
• $f(x)$ is irreducible $\bmod p$ iff the Frobenius element is a $3$-cycle iff $a_p = -1$.

For $p < 23$ the coefficients are the same as the coefficients of $q \prod_{n=1}^{\infty} (1 - q^n)$ which is $q$ times the Euler function, whose coefficients are given by the pentagonal number theorem. This gives that the $q$-expansion of $A$ begins

$$A(q) = q - q^2 - q^3 + q^6 + q^8 - q^{13} - q^{16} + \dots$$

hence

• $a_2 = -1$, meaning $x^3 - x - 1 \bmod 2$ is irreducible (which is true since it has no roots),
• $a_3 = -1$, meaning $x^3 - x - 1 \bmod 3$ is irreducible (which is true since it's a nontrivial Artin-Schreier polynomial)
• $a_5 = 0$, meaning $x^3 - x - 1 \bmod 5$ splits into a linear and a quadratic factor (given by $(x - 2)(x^2 + 2x - 2)$)
• $a_7 = 0$, meaning $x^3 - x - 1 \bmod 7$ splits into a linear and a quadratic factor (given by $(x + 2)(x^2 - 2x + 3)$)

and so forth. Apparently the smallest split prime is $p = 59$.

This MO question might also be relevant.

Shimura's article "A reciprocity law in non-solvable extensions" may be an example what you are looking for.