(i) In each of the following two cases, determine a highest common factor in $\mathbb{Z}[i]$ :

(a) $3+4 i, 4-3 i$;

(b) $3+4 i, 1+2 i$.

(ii) State and prove the Eisenstein criterion for irreducibility of polynomials with integer coefficients. Show that, if $p$ is prime, the polynomial

$$1+x+\cdots+x^{p-1}$$

is irreducible over $\mathbb{Z}$.