Thread: 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 $\displaystyle F_2$ of degree $\displaystyle \leq 3 $.

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

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys thanks.

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 $\displaystyle F_2$ of degree $\displaystyle \leq 3 $.

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

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys thanks.

what does $\displaystyle F_2$ mean here?

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 $\displaystyle F_2$ of degree $\displaystyle \leq 3 $.

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

(c) Factorise the reducible polynomials into irreducible polynomials.

any clues appreciated guys thanks.

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

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

We can think of this field as $\displaystyle \mathbb{Z}_2 \left< + , \cdot \right>$
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).