Suppose $E\_1$ and $E\_2$ are two elliptic curves defined over $\mathbb{Q}$ and there exists a 3-isogeny $\varphi$: $E\_1 \longrightarrow E\_2$. If $E\_1$ has no $\mathbb{Q}$-rational point of order 3, then what is Ker $\varphi$ $?$ Is it $\mu\_3$ or $\mathbb{Z}/3\mathbb{Z}$ where $\mu\_3$ denotes the 3rd roots of unity $?$



Kernel of a 3-isogeny between two elliptic curves
The simplest example is $y^2=x^3+D$, which has a 3-isogeny defined over $\mathbb{Q}$ whose kernel is $\{O,(0,\sqrt{D}),(0,-\sqrt{D})\}$. As abx pointed out, as a Galois module this kernel is defined by the quadratic extension $\mathbb{Q}(\sqrt{D})$, which can be any quadratic extension. In general, if the kernel is $\{0,P,-P\}$, then Galios acts via $\sigma(P)=\chi(\sigma)P$, and $\chi$ is the quadratic character in abx's comment. 


