When working with fields, it's a usual method to work on an algebraic closure of a field to obtain results about that field. In general (i. e. unless you're explicitly considering "well-behaved" fields like $\Bbb R$ or $ \mathbb{F}_q$), the existence of this algebraic closure is only provided by the Lemma of Zorn. Thus, proofs in field theory using the algebraic closure are not valid if you don't believe in the axiom of choice.
On the other hand, algebraic closures often are used to simplify the proof, but you could also get along without them. But are there any important cases where one is dependent on the existence of algebraic closures? In other words: Do you know theorems involving fields, which can only be proved if you use the existence of algebraic closures?