Question: Given,

$U\_t-U\_{xx}-2U\_x=0$

Using method of separation of variables, find ALL possible solution.

My answer :

$U(x,t)=X(x)T(t)$

$T'(t)/T(t)$=$[X''(x)+2X'(x)]/X(x)$=$\lambda$

From here I'm stuck. Could someone show me some working steps for me to proceed further.

Answer: 
Consider the splitting field of $x^5-1$, $x^2-2$ on $\mathbb{Q}$.
Actually, the simplest example is $\Bbb Q(\zeta\_{15})$, since cyclotomic extensions are well-known.

A quick calculation shows $\operatorname{Gal}(\Bbb Q(\zeta\_{15})/\Bbb Q) = (\Bbb Z/15\Bbb Z)^\times = (\Bbb Z/3\Bbb Z)^\times \times (\Bbb Z/5\Bbb Z)^\times = C\_2 \times C\_4$.