Carlo Wood on Mon, 28 Apr 2008 15:24:03 +0200

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 How to check irreducibility over a finite field?

```Hi,

I'd like to check the irredicubility of a given
polynomial over a field K.

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?

--
Carlo Wood <carlo@alinoe.com>

```