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This is one of the theorems in Field Arithmetic written by M. Fried and M. Jarden as following which is Proposition 13.4.6 in that monograph: Every field K has a regular extension F which is PAC ...

The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...