This problem deals with polynomials in Z_{2}\left[X\right]. Remember these

a) Suppose that any polynomial ${\displaystyle \uparrow p\left(x\right)}$ in ${\displaystyle Z_2\left[X\right]}$ has even number of nonzero coefficient then ${\displaystyle p\left(1\right)}$ is even (because the coefficient are eithes 0 or 1. So nonzero coefficient sum of coefficient which are even) and ${\displaystyle p\left(r\right)}$ is even ${\displaystyle \to 1}$ is root of ${\displaystyle p\left(x\right)}$ with operation ${\displaystyle \cdot \text{mod}Z\to }$ If 1 is root of ${\displaystyle p\left(x\right)\to p\left(x\right)}$ is reducible. So, if a polynomial is irreducible with a constant term than the Hamming weight (No of nonzero coefficient) is add.
(If constant term is not included then ${\displaystyle x=0}$ is a root andpolynomial is reducible)
b) We want a polynomial of degree Z with constant term which is irreducible ${\displaystyle i-e}$ no of no of nonzero coefficients is add. Since polynomial of degree with constant term has at last two nonzero coefficient (one coefficient of ${\displaystyle x^2}$ and 2nd constant term). Se, for total coefficient to be add coefficient of x must be nonzero. So ${\displaystyle f\left(x\right)=x^2+x+1=f\left(x\right)}$.
Under mod f operation any polynomial has degree at most 1. So, polynomials under mod f are
${\displaystyle \{1,x,1+x,0\}}$
$$\begin{array}{|ccccc|}\hline x& 0& 1& x& 1+x\\ 0& 0& 0& 0& 0\\ 1& 0& 1& x& 1+x\\ x& 0& x& 1+x& 1\\ 1+x& 0& 1+x& 1& x\\ \hline\end{array}$$
$$\begin{array}{|ccccc|}\hline +& 0& 1& x& 1+x\\ 0& 0& 1& x& 1+x\\ 1& 1& 0& 1+x& x\\ x& x& 1+x& 0& 1\\ 1+x& 1+x& x& 1& 0\\ \hline\end{array}$$ c) Irreducible polynomials of degree 3.
No of nonzero coefficient "with constant term" are add in an ${\displaystyle \downarrow }$ irreducible polynomial with constant term.
So, For degree 3, irreducible polynomial ${\displaystyle x^4+x^3+1}$, ${\displaystyle x^4+x^2+1}$ (For a poiynomunal of degree with constant term $x^3$ and constants has coefficient 1: So remainingterms ${\displaystyle x^2}$ and x; For nonzero coefficient to be add we can add any one of ${\displaystyle x^2}$ and x)
For polynomial of degree 4, polynomial with add no of nonzero coefficients and constant terms are
${\displaystyle x^4+x^3+1}$, ${\displaystyle x^4+x^2+1}$, ${\displaystyle x^4+x+1}$, ${\displaystyle x^4+x^3+x^2+x+1}$
Now, if any polynomial ${\displaystyle f\left(x\right)}$ of degree 4 can be factorized into ${\displaystyle f_1\left(x\right)}$ and ${\displaystyle f_2\left(x\right)}$ of degree 2, and ${\displaystyle f\left(x\right)}$ has hamming no add.
(If hamming no is even then ${\displaystyle f\left(x\right)}$ is reducible.)
Now ${\displaystyle f\left(x\right)}$ has hamming no add. So ${\displaystyle f_1\left(x\right)}$ and ${\displaystyle f_2\left(x\right)}$ also has hamming no add. Since any of ${\displaystyle f_1\left(x\right)}$, ${\displaystyle f_2\left(x\right)}$ has hamming no even then 1 is root of ${\displaystyle f_1\left(x\right)}$ o r${\displaystyle f_2\left(x\right)\to 1}$ is root of