Electron mobility in short and long channel, strained or unstrained FD-SOI MOSFETs is deeply investigated in linear regime, by careful magnetoresistance measurements down to 40 nm gate length, and down to 20 K. This method differs from standard ones because i) it does not require any data on the short channel gate capacitance and gate length; ii) it is more accurate at low inversion charge; iii) the temperature dependence of the Coulomb Scattering limited mobility is higher. Additional mobility scattering has been thus confirmed for short channel undoped FDSOI, and unambiguously identified as Coulomb scattering (CS). A 50% mobility gain for strained Si MOSFETs is still observable even in this dominant CS regime.