# listing polynomials?

• Oct 22nd 2007, 06:39 AM
smoothman
listing polynomials?
hi another one guys:

any clues on how to go about doing these questions would be greatly appreciated :) i will show my working once i have an idea :)

(a) list all the monic polynomials over $F_2$ of degree $\leq 3$.

(b) determine which of these polynomials are irreducible over $F_2$.

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys :) thanks.
• Oct 22nd 2007, 09:29 AM
Jhevon
Quote:

Originally Posted by smoothman
hi another one guys:

any clues on how to go about doing these questions would be greatly appreciated :) i will show my working once i have an idea :)

(a) list all the monic polynomials over $F_2$ of degree $\leq 3$.

(b) determine which of these polynomials are irreducible over $F_2$.

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys :) thanks.

what does $F_2$ mean here?
• Oct 22nd 2007, 10:51 AM
ThePerfectHacker
Quote:

Originally Posted by smoothman
hi another one guys:

any clues on how to go about doing these questions would be greatly appreciated :) i will show my working once i have an idea :)

(a) list all the monic polynomials over $F_2$ of degree $\leq 3$.

(b) determine which of these polynomials are irreducible over $F_2$.

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys :) thanks.

To Jhevon: Here $\mathbb{F}_2$ is the field that has two elements (another notation $\mbox{GF}(2)$ which means "Galois field").

And degree <=3 polynomial is,
$Ax^3+Bx^2+Cx+D$ where $A,B,C,D\in \mathbb{F}_2$.

We can think of this field as $\mathbb{Z}_2 \left< + , \cdot \right>$
So the elements are:
x^3,x^3+x^2,x^3+x,x^3+1,x^3+1,x^3+x^2+1,x^3+x^2+x, x^3+x^2+1.x^3+x^2+x+1,x^2,x^2+x,x^2+1,x^2+x+1,x,x+ 1

Now to see which are irreducible over this field simply check if it has a zero or not (because it is of degree 3 or less).