what is voltage across an inductor?
In DC, it behaves as a linear devices (for ideal cases, nonlinear when it comes to magnetostatics), so the voltage across an inductor is \$V=L\frac{di}{dt}\$. Things get nonlinear when there is resistance in the same circuit (e.g. an RL Circuit).

In AC, the impedance relies on the frequency of the voltage going through it, thus in steady state, the impedance can be characterized as \$Z_L = j\omega L\$. Therefore, the voltage across the inductor would be \$V=IX_L\$, where \$X_L\$ would just be \$Im\{Z_L\}\$ or \$\omega L\$. If you were to talk about the instantaneous voltage across an inductor, you would get: \$V(t) = V_{RMS}\sin (\omega t)\$. The voltage also leads the current by \$90 ^{\circ}\$.

You asked about emf. For emf, the equation is \$V=-L\frac{di}{dt}\$. I'm not going to delve into Faraday's Law because there's no reason to talk about that math and things get more complicated in AC. But if you're curious, go to this question if you want to learn about why there's confusion. Sergei's answer is actually a very good explanation.