Bill Allombert on Mon, 28 Apr 2008 18:47:31 +0200

 Re: How to check irreducibility over a finite field?

```On Mon, Apr 28, 2008 at 03:23:40PM +0200, Carlo Wood wrote:
> Hi,
>
> I'd like to check the irredicubility of a given
> polynomial over a field K.

Please note that there is a function polisirreducible for that purpose.

> Let K = F_2[t]/r(t)F_2[t], where r(t) is a given
> reduction polynomial over F_2.
>
> Let L = K[s]/u(s)K[s], where u(s) is a given
> reduction polynomial over K.
>
> I want to manually pick a u(s) and then check
> if I can factor it in K.
>
> Here is what I tried:
>
> First I define r(t):
>
> ? r = Mod(1,2)*t^6 + Mod(1,2)*t + Mod(1,2);
>
> It is possible to check that this is irreducible over F_2 with:
>
> ? lift(factormod(r, 2))
> %2 =
> [t^6 + t + 1 1]
>
> Next, I define a u(s):
>
> ? u = Mod(Mod(1,2),r)*s^2 + Mod(Mod(1,2),r)*s + Mod(Mod(1,2),r);
>
> Now I want to see if I can factor this over K:
>
> ? lift(factormod(u, r))
>   *** factormod: incorrect type in factmod.
>
> Not even this works:
>
> ? lift(factormod(u, 2))
>   *** factormod: incorrect type in factmod.
>
> How can I factor a polynomial u(s) over K?

Use factor(u) or factorff(u,2,r) if the moduli are implicit.

Cheers,
Bill.

```