I am not sure how predicate logic would give you an algorithm for deciding if two expressions are equal. It seems that you need to decide if an equation follows from field axioms. The method that comes to mind is to normalize both sides and see if they are identical. Let's say you only deal with ring operations, i.e., polynomials. First, you need to order the variable names, for example, using alphabetical order. Then the standard form of a monomial would be where the sequence is ordered. Then you need to decide how to order monomials, probably again using some version of alphabetic order. After that, every expression should have a single normal form, which is a sum of monomials (possibly with coefficients) in the given order. Given two expressions, you can normalize them and compare their normal forms.
There may be smarter ways to define the normal form, for example, using Horner form. You may want to look at the paper "Proving Equalities in a Commutative Ring Done Right in Coq" by Benjamin Grégoire and Assia Mahboubi. It is pretty advanced, but you only need to get the idea and some implementation details.
Of course, dealing with fields is more complicated. Overall, I am wondering if it is easier to employ some computer algebra program and write a script that would ship an equation to it and return the result.