My mother found a hornet at home (in northern France) that is probably building a nest. Is that an asian hornet? If yes, how can I exterminate them?
This is not a hornet, this a wasp. See the yellow dots on the thorax ? They are typical of the Vespula genus. So is the yellow and black stripe pattern on the abdomen. Those wasps are smaller than hornets (1-2 cm for wasps, 3-4 cm for hornets). Picture from Wikipedia:
There are two common Vespula species in France (V. vulgaris and V. germanica), but to identify yours, we would need a better view of the head and abdomen.
On the image below, you can compare it to the two hornet species you can find in France (source). The first one is a bee, the second one is a wasp (Vespula), the third one is the invasive "Asian hornet" (Vespa velutina) and the fourth one is the native "European hornet" (Vespa crabro).
So no extermination needed ;) You can find more information (in French) on how to recognize these species on the Natural History Museum website.