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A set of multivariate polynomials is algebraically independent if they exhibit no non-trivial algebraic relations, and this notion is fundamental in algebra. When these polynomials are given as algebraic circuits, deciding algebraic independence has several applications in algebraic complexity theory. Over fields of zero (or exponentially large) characteristic, this problem is known to have an efficient randomized algorithm. Over finite fields of small characteristic, a sequence of works has culminated in showing that algebraic independence admits Arthur-Merlin proofs, in particular giving the complexity bound of AM∩coAM ([Guo et al., 2019]). We improve the complexity of deciding algebraic independence over finite fields by showing that it admits zero-knowledge proofs, in particular giving the upper bound of NISZK ⊆ AM∩coAM, the class of problems admitting non-interactive statistical zero-knowledge proofs. This is achieved by arguing that algebraically independent polynomials yield maps whose output distribution has high-entropy, while algebraically dependent polynomials yield maps with low-entropy. We can then reduce to the question of approximating entropy, which is a known NISZK-complete problem. We also more generally show that transcendence degree, which quantifies the independence of a set of possibly dependent polynomials, can be computed in NISZK.