= Solution
For $d=1$ and $X\sim N(0,\sigma^2)$, the <characteristic function> is
$$
\boxed{\varphi_X(u)=e^{-\sigma^2u^2/2}}.
$$
For a <multivariate normal distribution> with covariance matrix $\sigma^2I_d$, the coordinates are uncorrelated and hence independent by <independence of uncorrelated jointly normal variables>. Therefore
$$
\varphi_X(u)
=\prod_{j=1}^d\mathbb E[e^{iu_jX_j}]
=\prod_{j=1}^de^{-\sigma^2u_j^2/2}
=\boxed{\exp\left(-\frac{\sigma^2}{2}\sum_{j=1}^du_j^2\right)}.
$$
This is the <characteristic function of an isotropic Gaussian vector>.
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