Homework Soursea Find an irreducible polynomial px of degree 2 in 2x If is
a) Find an irreducible polynomial p(x) of degree 2 in 2[x].
If is a zero of p(x) of part a) in some extension field of 2, find how many elements the field Z2[x]/p(x) has
Factor p(x) of part a) 2()[x].
Solution
a) The polynomial P(x) = x^2 + 1, is an irreducible polynomial in Z2[x], since the roots of the polynomial P(x) are x=i and x=-i
Therefore, the polynomial doesn\'t have any root of degree 2 in the domain of Z2[x]
b) If considering alpha has a zero, then the extension field must contain i and -i as well,hence the factors of Z2[x]/p(x) will be equal to
(x+i) and (x-i)
![a) Find an irreducible polynomial p(x) of degree 2 in 2[x]. If is a zero of p(x) of part a) in some extension field of 2, find how many elements the field Z2[x]](/WebImages/15/a-find-an-irreducible-polynomial-px-of-degree-2-in-2x-if-is-1024733-1761530399-0.webp "a) Find an irreducible polynomial p(x) of degree 2 in 2[x]. If is a zero of p(x) of part a) in some extension field of 2, find how many elements the field Z2[x]")
