I'm not sure why you say this.  I am right now looking at the following paper of Ax:

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MR0229613 (37 #5187)
Ax, James
The elementary theory of finite fields.
Ann. of Math. (2) 88 1968 239--271. 

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The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields."  He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q.  He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!). 

The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH.  This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.