I've seen this stated on this site as well as elsewhere that there is no free object in the category of fields on a set $X$ of any size including the empty set. I'm wondering why this is the case?
2026-04-07 07:49:08.1775548148
Free object in the category of fields
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Hint: Assume $F$ is a free field on the set $X$. Then try to extend any map $X\to K$ to a homomorphism where $K$ has different characteristic.