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I don't have a proof as much as a fairly weak (in my opinion) arguement.

Suppose you had a series of functions with coefficients p. If the coeffients are p in Zp[x], all those functions go to zero. In this case you just have a bunch of zeroes, which can't happen since there is no zero associate in Zp[x]. I think I may have butchered this proof as we just went over associates and I'm not sure how to do this.