When such computation is fast, it depends on some optimized library (note that Sage can be seen as a bundle of optimized libraries behind a uniform Python interface), and that one depends on the type of polynomials. If you do:

```
sage: R.<x> = QQ[]
sage: P = R.random_element()
sage: P.is_irreducible??
```

You will see that `flint`

is used.

If you do the same by replacing `QQ`

with `GF(2)`

, you will see that the `gf2x`

library is used but for `GF(p)`

with p>2, the `nmod_poly_is_irreducible`

function is used and that one relies on `flint`

as well. If you have polynomials over `ZZ`

, then you will see that the `factor`

method is used, and there you will see that `pari`

is used when the degree of the polyomial is between 30 and 300, and `ntl`

is used otherwise, and so on.

So, you have to see the source of the corresponding upstream program to see which algorithm and optimizations they are using.

Polynomial in one variable? Over which ring?

A univariate polynomial over the integers

In this case, a mixture between ntl and pari is used as i explained.