Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?



bad reduction for elliptic curves
Here are two answers:


(a) If you try to write down an elliptic curve $y^2 = x^3 + a x + b$ with everywhere good reduction, you need to choose $a$ and $b$ such that $4a^3 + 27 b^2 = $ a unit. We can certainly solve this equation over some (lots!) of number fields, say if we set the unit equal to $1$ or $-1$, or a unit in some fixed base number field. But we can't solve it in ${\mathbb Q}$. 


[Edit: As Bjorn intimates in his comment below, one has to be a little more careful than I am being here to be sure of good reduction mod primes above 2; the details are left to the interested reader (or, I imagine, can be found in Silverman in the section where he discusses the proof that there are no good reduction elliptic curves over $\mathbb Q$).]


(b) There are many non-trivial everywhere unramified extensions of number fields (e.g. $\mathbb Q(\sqrt{-5}, i)$ over $\mathbb Q(\sqrt{-5})$), but there are no everywhere unramified extensions of the particular number field $\mathbb Q$. The situation with elliptic curves is completely analogous.


