A permanent magnet rotating electric machine for a vehicle, the machine comprising a stator, a rotor arranged rotatably around a rotation axis inside said stator, and a magnetic pole position detector, wherein said stator includes: a stator core having plural slots in an inner surface side thereof, each slot extending in a direction of the rotation axis, and the plural slots being adjacent to each other in a circumferential direction, and a stator winding built as three-phase distributed winding and accommodated in the slots of said stator core; said rotor includes: a rotor core in which a plurality of poles are equi-angularly formed, and a plurality of permanent magnets arranged in said rotor core to form said plurality of poles; characterized in that said rotor core includes a first air gap located in the center portion of each of said plurality of poles wherein a width of the air gap in a radial direction at the center of said air gap is larger than the radial width of a portion of said air gap circumferentially distant from the center, said rotor core includes per pole two permanent-magnet-inserting holes (34AX, 34CX) accommodating permanent magnets, said two permanent-magnet-inserting holes (34AX, 34CX) being formed symmetrically with respect to a symmetry plane passing through the center of the pole and accommodating the rotation axis, formed to have a cross-sectional shape of a concave arc, seen from the symmetry plane, and such that hole portions nearer to the rotation axis are mutually nearer and extend in the direction of the rotation axis, wherein permanent magnets of adjacent poles are magnetized in reverse polarity, wherein permanent magnets are inserted into the left half of the axial length of the permanent-magnet insertion holes (34Ax, 34Cx) and into the right half of a second air gap, said rotor core includes a second air gap formed symmetrically with respect to the symmetry plane and between rotation-axis-side ends of said two permanent-magnet-inserting holes (34AX, 34CX) so as to have a cross-sectional shape of a concave arc. a stator core having plural slots in an inner surface side thereof, each slot extending in a direction of the rotation axis, and the plural slots being adjacent to each other in a circumferential direction, and a stator winding built as three-phase distributed winding and accommodated in the slots of said stator core; said rotor includes: a rotor core in which a plurality of poles are equi-angularly formed, and a plurality of permanent magnets arranged in said rotor core to form said plurality of poles; characterized in that said rotor core includes a first air gap located in the center portion of each of said plurality of poles wherein a width of the air gap in a radial direction at the center of said air gap is larger than the radial width of a portion of said air gap circumferentially distant from the center, said rotor core includes per pole two permanent-magnet-inserting holes (34AX, 34CX) accommodating permanent magnets, said two permanent-magnet-inserting holes (34AX, 34CX) being formed symmetrically with respect to a symmetry plane passing through the center of the pole and accommodating the rotation axis, formed to have a cross-sectional shape of a concave arc, seen from the symmetry plane, and such that hole portions nearer to the rotation axis are mutually nearer and extend in the direction of the rotation axis, wherein permanent magnets of adjacent poles are magnetized in reverse polarity, wherein permanent magnets are inserted into the left half of the axial length of the permanent-magnet insertion holes (34Ax, 34Cx) and into the right half of a second air gap, said rotor core includes a second air gap formed symmetrically with respect to the symmetry plane and between rotation-axis-side ends of said two permanent-magnet-inserting holes (34AX, 34CX) so as to have a cross-sectional shape of a concave arc. a rotor core in which a plurality of poles are equi-angularly formed, and a plurality of permanent magnets arranged in said rotor core to form said plurality of poles; characterized in that said rotor core includes a first air gap located in the center portion of each of said plurality of poles wherein a width of the air gap in a radial direction at the center of said air gap is larger than the radial width of a portion of said air gap circumferentially distant from the center, said rotor core includes per pole two permanent-magnet-inserting holes (34AX, 34CX) accommodating permanent magnets, said two permanent-magnet-inserting holes (34AX, 34CX) being formed symmetrically with respect to a symmetry plane passing through the center of the pole and accommodating the rotation axis, formed to have a cross-sectional shape of a concave arc, seen from the symmetry plane, and such that hole portions nearer to the rotation axis are mutually nearer and extend in the direction of the rotation axis, wherein permanent magnets of adjacent poles are magnetized in reverse polarity, wherein permanent magnets are inserted into the left half of the axial length of the permanent-magnet insertion holes (34Ax, 34Cx) and into the right half of a second air gap, said rotor core includes a second air gap formed symmetrically with respect to the symmetry plane and between rotation-axis-side ends of said two permanent-magnet-inserting holes (34AX, 34CX) so as to have a cross-sectional shape of a concave arc.