On the reducibility of $x^4+ a$ over the rationals

Question

Let $a \in \mathbb{Z^{+}}$. Show that $x^4+a$ is reducible over $\mathbb{Q}$ if and only if $a=4b^4$ for some integer $b$.

My idea for one implication was to assume reducibility and write $x^4+a = (x^2+\alpha_1x + \beta_1)(x^2+\alpha_2x + \beta_2)$, leading to the system of equations: $\alpha_1 + \alpha_2 = 0$, $\beta_1+\beta_2+\alpha_1 \alpha_2 = 0$, $\alpha_2 \beta_1 + \alpha_1 \beta_2 = 0$, $\beta_1 \beta_2 = a$. But I wasn't able to get much farther than this. This is a practice problem for an exam, any help is appreciated.

Answer 1

You don't solve the whole system at once. Since the cubic coefficient must be zero, we must have $$ (x^2 + cx + d)(x^2 - cx + e) $$ The linear coefficient is $ce - cd,$ and this must be zero. So, $c(d-e) = 0.$ When $c=0,$ in order to get the quadratic coefficient zero, we need $d+e = 0.$ So the case $c=0$ gives $$ x^4 -d^2, $$ which is ruled out

The other case is $d=e.$ We continue with $$ (x^2 + cx + d)(x^2 - cx + d) = x^4 + (2d-c^2)x^2 + d^2 $$ Since $c,d$ are integers, we have $c = 2 \gamma$ and $d = 2 \gamma^2,$ finally $$ (x^2 + 2 \gamma x + 2 \gamma^2) (x^2 - 2 \gamma x + 2 \gamma^2) $$ which becomes..........

Answer 2

Another solution is to first factor out into linear factors over $\mathbb{C}$, and multiply conjugate factors to obtain the real quadratic factors:

$$X^4 + a = \left(X^2 + ia^{1/2}\right)\left(X^2 - ia^{1/2}\right) = \color{red}{\left(X + \frac{1-i}{\sqrt{2}}a^{1/4}\right)}\color{blue}{\left(X - \frac{1-i}{\sqrt{2}}a^{1/4}\right)}\color{red}{\left(X + \frac{1+i}{\sqrt{2}}a^{1/4}\right)}\color{blue}{\left(X - \frac{1+i}{\sqrt{2}}a^{1/4}\right)}$$ $$ = \color{red}{\left(X^2 + \sqrt{2}a^{1/4}X + a^{1/2}\right)}\color{blue}{\left(X^2 - \sqrt{2}a^{1/4}X + a^{1/2}\right)}$$

where we used $\sqrt{i} = \pm \frac{1+i}{\sqrt{2}}$ and $\sqrt{-i} = \pm\frac{1-i}{\sqrt{2}}$.

(you can also check the factorization directly without using complex numbers).

From this expression it is clear that we need $\sqrt{2}a^{1/4} = 2r$ for some $r \in \mathbb{Q}$ which implies $a = 4r^4$.