Electric machine having a fluid cooling device , comprising - a stator having a stator laminated core and having a stator winding , - a rotor having a rotor laminated core , said rotor being mounted so as to be rotatable about an axis (A) relative to the stator by means of a rotor shaft and a first bearing which is arranged on a first end shield , and wherein - the fluid cooling device has a cooling tube which is fixed to a carrier element , which is fixed relative to the stator, of the electric machine and is connected to a fluid inlet and which extends axially within a central cutout of the rotor shaft and so as to form an annular space relative to the rotor shaft , wherein - the cooling tube is fluidically connected to the annular space and to a fluid outlet , and wherein - a sealing region having a sealing element is provided between the rotor shaft and the carrier element , said sealing element at least substantially preventing a transfer of fluid into a space region of the electric machine that lies outside the sealing region when a cooling fluid is flowing through from the fluid inlet to the fluid outlet , wherein - for the mounting of the rotor , a second bearing which is axially spaced apart from the first bearing is provided between the rotor shaft and a second end shield , wherein the rotor laminated core extends in a space between the first and the second bearing , characterized in that - the rotor shaft has a shaft portion which protrudes axially beyond the second bearing and has an output element , wherein the central cutout of the rotor shaft extends axially into the region of the output element and is flowed through by the cooling fluid. - a stator having a stator laminated core and having a stator winding , - a rotor having a rotor laminated core , said rotor being mounted so as to be rotatable about an axis (A) relative to the stator by means of a rotor shaft and a first bearing which is arranged on a first end shield , and wherein - the fluid cooling device has a cooling tube which is fixed to a carrier element , which is fixed relative to the stator, of the electric machine and is connected to a fluid inlet and which extends axially within a central cutout of the rotor shaft and so as to form an annular space relative to the rotor shaft , wherein - the cooling tube is fluidically connected to the annular space and to a fluid outlet , and wherein - a sealing region having a sealing element is provided between the rotor shaft and the carrier element , said sealing element at least substantially preventing a transfer of fluid into a space region of the electric machine that lies outside the sealing region when a cooling fluid is flowing through from the fluid inlet to the fluid outlet , wherein - for the mounting of the rotor , a second bearing which is axially spaced apart from the first bearing is provided between the rotor shaft and a second end shield , wherein the rotor laminated core extends in a space between the first and the second bearing , characterized in that - the rotor shaft has a shaft portion which protrudes axially beyond the second bearing and has an output element , wherein the central cutout of the rotor shaft extends axially into the region of the output element and is flowed through by the cooling fluid. <EOS>