Key to "List all of the polynomials of degrees 2..."

Anish Buchanan

Answered question

2021-09-18

List all of the polynomials of degrees 2 and 3 in

Z_{2} [x]

. Find all of the irreducible polynomials of degrees 2 and 3 in

Z_{2} [x]

Answer & Explanation

coffentw

Skilled2021-09-19Added 103 answers

Let $a x^{3} + b x^{2} + c x + d \in Z_{2} [x]$ be a polynomial of degree 3.
then we must have a=1
for this polynomial to be irreduicble we must also d=1
since otherwise we will have a polynomial $x^{3} + b x^{2} + c x = x (x^{2} + b x + c)$
that can be factored an therefore reducible.
so the polynomial $f (x) = x^{3} + b x^{2} + c x + 1$ , where b, $c \in Z_{2} [x]$
so the polynomial has no zero in $Z_{2} [x]$
so, we may not have $f (0) = 0$ of $f (1) = 0$
now f(0)=1 no matter chosen $f (1) = b + c$
hence we have to choose b and c so, that $b + c = 1$
and the two possibilities $b = 1, c = 0 or b = 0, c = 1$ .
that there are two irreducible polynomials of degree $3 \in$ $Z_{2} [x]$
$f_{1} (x) = x^{3} + x^{2} + 1$
$f_{2} (x) = x^{3} + x + 1$

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