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 of degree .
(b) determine which of these polynomials are irreducible over .
(c) Factorise the reducible polynomials into irreducible polynomials.
any clues appreciated guys thanks.
To Jhevon: Here is the field that has two elements (another notation which means "Galois field").
And degree <=3 polynomial is,
where .
We can think of this field as
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).