It would be quite a shock for the French to allow such a prestigious French company to be taken over by an America company, the TGV manufacturer being owned by the Yanks, Sacre Bleu!
As has been explained, it is not really a French company if you look at history.
Historically Alstom IS French GE.
Alstom came about exactly because of GE in the late 19th / early 20th centuries.
Today the name Alstom means nothing, that is just brand mark.
Before Alstom was Alsthom - with an h.
Als meant Alsace-Lorraine.
Thom meant Thomsom-Houston.
Thomson-Heuston was a US electrical concern in 1892 merged with (US) GE. GE kept on the TH name for certain parts of ths business including UK British Thomson Houston and 1893 formed Compagnie Francaise Thomson-Houston (CFTH).
Lots of mergers takeover splits spinoffs etc by 1920/1930 GE had relinquised control of these companies, leaving, for the purposes of this thread, AEI in UK (BTH + MV) and Alsthom in France.
Digressing, GEC - UK GEC that is - had nothing whatsover to do with any of this until Weinstocks late 1960s moves by UK GEC (never linked with GE) taking over AEI etc and further takeover and mergers leading to GEC Alsthom, subsequently unravelled when GEC broke up post-Weinstock.
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Nick