Are $(X−λ),(X^3),(λ),(X^2−1)$ prime and maximal ideal in K[X]?

Question

(i) $(X−λ),(X^3),(λ),(X^2−1)$ in K[X], where K is a field and and λ∈K∖{0}λ∈K∖{0}.
(ii) $(13);(2,X^3+X^2+X+1)$in $\mathbb{Z}[X]$.
($(2,X^3+X^2+X+1)$ is an ideal generated by 2 and $X^3+X^2+X+1$

I have no idea on how to show which ideal is a prime or maximal. Please show me some hints? many thanks

Answer 1

If K is R, $(X^2-1)$ is not prime since $X^2-1 = (X+1)(X-1)$

$(X^3)$ is not prime since $X^3 =XX^2$.

If $K$ is any field, $P,Q$ in $K[X]$, $PQ\in (X-c), c\neq 0$ implies that $PQ=R(X-c)$. This implies that $P(c)Q(c) =0$, since $K$ is a field, $P(c) =0$ or $Q(c)=0$. If $P(c)=0$, divides $P$ by $(X-c)$, $P=U(X-c)+d, d\in K$. $P(c)=d=0$ implies that $P=U(X-c)$. Thus $(X-c)$ is prime.