A rotary electric machine rotor comprising: a Lundell core that comprises: a cylindrical portion (5, 9); a pair of ring-shaped yoke portions (6, 10) that are disposed on two axial end portions of said cylindrical portion (5, 9); and a plurality of claw-shaped magnetic pole portions (7, 11) that are disposed so as to extend axially alternately from each of said pair of yoke portions (6, 10), said claw-shaped magnetic pole portions (7, 11) being arranged circumferentially so as to intermesh with each other; a rotating shaft that is inserted into and fixed to a central axial position of said Lundell core; a bobbin that comprises: an annular drum portion that is mounted around an outer circumference of said cylindrical portion (5, 9); and a pair of flat ring-shaped flange portions that protrude radially outward from two axial end portions of said drum portion; a field coil (12, 12A, 12B) that is wound in multiple layers in a space that is surrounded by said drum portion and said pair of flange portions, said field coil (12, 12A, 12B) contacting a vicinity of roots of inner circumferential surfaces of said claw-shaped magnetic pole portions (7, 11); and a cooling fan (14, 15, 16) that supplies air to an outer circumferential surface of said field coil (12, 12A, 12B), wherein said field coil (12, 12A, 12B) is formed so as to have a peaked shape in which two or more peaks line up consecutively in an axial direction, characterized in that an apex portion of each of said peaks is positioned radially further outward than an outside diameter of a root position of said claw-shaped magnetic pole portions (7, 11), wherein a floor portion of a valley that is positioned between peaks that are adjacent to each other in an axial direction of said field coil (12A, 12B) is positioned radially further inward than an outside diameter of a root position of said claw-shaped magnetic pole portions (7, 11). a Lundell core that comprises: a cylindrical portion (5, 9); a pair of ring-shaped yoke portions (6, 10) that are disposed on two axial end portions of said cylindrical portion (5, 9); and a plurality of claw-shaped magnetic pole portions (7, 11) that are disposed so as to extend axially alternately from each of said pair of yoke portions (6, 10), said claw-shaped magnetic pole portions (7, 11) being arranged circumferentially so as to intermesh with each other; a cylindrical portion (5, 9); a pair of ring-shaped yoke portions (6, 10) that are disposed on two axial end portions of said cylindrical portion (5, 9); and a plurality of claw-shaped magnetic pole portions (7, 11) that are disposed so as to extend axially alternately from each of said pair of yoke portions (6, 10), said claw-shaped magnetic pole portions (7, 11) being arranged circumferentially so as to intermesh with each other; a rotating shaft that is inserted into and fixed to a central axial position of said Lundell core; a bobbin that comprises: an annular drum portion that is mounted around an outer circumference of said cylindrical portion (5, 9); and a pair of flat ring-shaped flange portions that protrude radially outward from two axial end portions of said drum portion; an annular drum portion that is mounted around an outer circumference of said cylindrical portion (5, 9); and a pair of flat ring-shaped flange portions that protrude radially outward from two axial end portions of said drum portion; a field coil (12, 12A, 12B) that is wound in multiple layers in a space that is surrounded by said drum portion and said pair of flange portions, said field coil (12, 12A, 12B) contacting a vicinity of roots of inner circumferential surfaces of said claw-shaped magnetic pole portions (7, 11); and a cooling fan (14, 15, 16) that supplies air to an outer circumferential surface of said field coil (12, 12A, 12B), wherein said field coil (12, 12A, 12B) is formed so as to have a peaked shape in which two or more peaks line up consecutively in an axial direction, characterized in that an apex portion of each of said peaks is positioned radially further outward than an outside diameter of a root position of said claw-shaped magnetic pole portions (7, 11), wherein a floor portion of a valley that is positioned between peaks that are adjacent to each other in an axial direction of said field coil (12A, 12B) is positioned radially further inward than an outside diameter of a root position of said claw-shaped magnetic pole portions (7, 11).