To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods.
What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture ? Can you suggest any new ideas involving algebraic methods or concepts that have been proposed to make progress on Goldbach's conjecture?
There are no major algebraic number theory approaches. These kinds of questions of additive number theory don't seem to admit any useful reinterpretation that lends itself to an algebraic approach (as far as I know).