On what refractive index of medium depends ? does the refractive index of a medium depends on wavelength of light passing through it ? If yes then why ? 
 A: Refractive index of a medium depends upon the refractive index of the surroundings (when you consider the light ray passing from the surrounding into the medium), optical density, wavelength of the light and temperature.
Yeah it does depend on the the wavelength of light, because:
As wavelength of light decreases, the velocity of light decreases.
Now, we know, absolute refractive index of a medium is equal to the (speed of light in air)/(speed of light in that medium), therefore if the velocity of light changes, the refractive index of the medium (for that wavelength of light) also changes. 
Example: it's due to this difference in velocities that dispersion of sunlight (to form rainbow) takes place. While rainbow is formed, the different colors (with different wavelengths) bend at different angles, but as all colors of light pass through same medium, thus, refractive index of medium does depend on the wavelengths.
A: Yes it depends on wave length it is inversely proportional to it. Let $u$ denote the refractive index, then:
$$u = c/v $$
where $c$ in the speed of light in vacuum and $v$ is the speed in a medium. And:
$$v=fl$$
where $f$ denotes the frequency and $l$ the wavelength. So we get:
$$ \frac{U_1}{U_2} = \frac{l_2}{l_1}$$
