Let $F$ be a field of characteristic $\neq 2$, and consider a polynomial $f(X) = X^4 + c_1X^2 + c_2 \in F[X]$.
(a) Use Problem 1 (ii) to show that $X^2 + aX + b$ is a factor of $f(X)$ if and only if $X^2 - aX + b$
is a factor of $f(X)$.
Show that if $f(X) = (X^2 + aX + b)(X^2 - aX + b)$, then $c_1 = 2b - a^2$ and $c_2 = b^2$.
(b) Show that $f(X) = (X^2 + b_1)(X^2 + b_2)$ if and only if $c_1^2 - 4c_2$ is a square in $F$.
(c) Use the above to show that $f(X) = X^4 + c \in F[X]$ is reducible if and only if either $-c$ is a
square in $F$ or $c = b^2$ is the square of an element $b \in F$ such that $2b$ is also the square of an
element of $F$.
In particular, show that $X^4 + 16$ is irreducible in $\mathbb{Q}[X]$, but that $X^4 + 4$ is reducible in $\mathbb{Q}[X]$,
and find an explicit factorization of $X^4 + 4$.
