WI: Prince Eugene of Savoy fights for the French Army

OTL Eugene applied directly to Louis XIV to join the French Army. What if Louis XIV says yes to him or what if Eugene applied to the army itself and got in? Basically what if Prince Eugene ended up as a French general instead of an Austrian one?
 
This may cause huge changes.

What happens against the Ottomans ? Of course, Austria had other good generals but would them win at Zenta or win such a great victory ?

And France may win decisive victories in the war of the league of Augsburg and the war of spanish succession.
 
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