A fixture for a hot or semi-hot forging machine including a system for applying a vertical compression force, movable in a vertical descent movement and in a vertical rising movement between an upper forging standby position and a lower post-forging position, the fixture being intended to forge gear teeth on an outer surface of revolution of a peripheral portion of a semi-finished product having an upper central portion extending opposite to a lower central portion, the fixture including: - a fixed lower punch provided to delimit a support for the lower central portion of the semi-finished product to be forged; - an upper counter-punch provided to have a clamping portion for the upper central portion of the semi-finished product, this counter-punch being mounted movable vertically to occupy a lower position for which the punch and the counter-punch immobilize the semi-finished product with its peripheral portion protruding relative to the punch and to the counter-punch and an upper position for which the counter-punch is disengaged relative to the semi-finished product to allow the placement of the semi-finished product and the extraction, relative to the punch, of the extruded semi-finished product; - a tool holder which the counter-punch passes through and supporting extrusion tooling appearing in the form of at least one ring provided, starting with its bore, with extrusion teeth provided with a forming ratio varying progressively to produce the gear teeth on the semi-finished product, the tool holder being provided to receive the vertical compression force of the application system of the forging machine so as to be moved at least vertically to extrude the gear teeth of the semi-finished product during the vertical descent movement of the system for applying the vertical compression force between the upper forging standby position for which the extrusion tooling is located above the support and the lower post-forging position characterized in that, in said lower post-forging position, the extrusion tooling is located entirely below the support and surrounding the punch. - a fixed lower punch provided to delimit a support for the lower central portion of the semi-finished product to be forged; - an upper counter-punch provided to have a clamping portion for the upper central portion of the semi-finished product, this counter-punch being mounted movable vertically to occupy a lower position for which the punch and the counter-punch immobilize the semi-finished product with its peripheral portion protruding relative to the punch and to the counter-punch and an upper position for which the counter-punch is disengaged relative to the semi-finished product to allow the placement of the semi-finished product and the extraction, relative to the punch, of the extruded semi-finished product; - a tool holder which the counter-punch passes through and supporting extrusion tooling appearing in the form of at least one ring provided, starting with its bore, with extrusion teeth provided with a forming ratio varying progressively to produce the gear teeth on the semi-finished product, the tool holder being provided to receive the vertical compression force of the application system of the forging machine so as to be moved at least vertically to extrude the gear teeth of the semi-finished product during the vertical descent movement of the system for applying the vertical compression force between the upper forging standby position for which the extrusion tooling is located above the support and the lower post-forging position characterized in that, in said lower post-forging position, the extrusion tooling is located entirely below the support and surrounding the punch.