By using Snell's law, $n_1\sin {\theta }_1=n_2\sin {\theta }_2$, the refractive index is 

$\mu =\frac{\sin 45°}{\sin 30°}=\frac{{\displaystyle \frac{1}{\sqrt{2}}}}{{\displaystyle \frac{1}{2}}}=\sqrt{2}$

Also using the relation $n=\frac{c}{v}$ where $n$ is the refractive index, $c$ is the velocity of light in air and $v$ is the velocity in the medium

$\Rightarrow \frac{c_{\text{air }}}{\text{v}}=\sqrt{2}=\frac{{\lambda }_{\text{1 }}}{{\lambda }_2}$

The velocities are given by 

$$\begin{gathered} v=\nu {\lambda }_2 \\ c=\nu {\lambda }_1 \end{gathered}$$

$\Rightarrow {\lambda }_1=\sqrt{2}{\lambda }_2$ and ${\nu }_1={\nu }_2$.

The frequency of light does not change when the medium changes.