My textbook, Introduction to Probability, first edition, by Blitzstein and Hwang, claims the following:
If $X_1$ and $X_2$ are independent and $X_1 + X_2$ is Normal, then $X_1$ and $X_2$ must be Normal! This is known as Cramér’s theorem. Proving this in full generality is difficult, but it becomes easier if $X_1$ and $X_2$ are i.i.d. with MGF $M(t)$. Without loss of generality, we assume $X_1 + X_2 \sim \text{N}(0,1)$, then the MGF of the sum is
$$e^\frac{t^2}{2} = E(e^{t(X_1 + X_2)}) = E(e^{tX_1})E(e^{tX_2}) = (M(t))^2,$$
so $M(t) = e^\frac{t^2}{4}$, which is the $\text{N}(0, 1/2)$ MGF. Thus, $X_1, X_2 \sim \text{N}(0, 1/2). \tag*{$\square$}$
At the end, the author has that $M(t) = e^\frac{t^2}{4}$. But the derived result was $e^\frac{t^2}{2} = (M(t))^2$. So shouldn't we have that $M(t) = e^\frac{t}{\sqrt{2}}$ and $X_1, X_2 \sim \text{N}(0, 1/\sqrt{2})$?
I would greatly appreciate it if people could please take the time to clarify this.