[zmgsabst]

My understanding is that you do your math over some field F.

But then when you choose a random point to test your polynomial, you randomly select from G = F[5^1/3], an extension of the original field. And test your polynomial using arithmetic in that larger field.

The increased entropy happens when you select at random from the extended field — there’s more elements in G than in F, so an attacker has a lower chance of guessing your random value.

[dlubarov]

Exactly - we sample uniformly from an extension field, so entropy is proportional to the extension field size. The base field is almost irrelevant from a security perspective, since things like the Schwartz-Zippel lemma just care about the size of the field we sample from, even if the polynomial in question is (also) a polynomial over some smaller subfield.