Unsolved Problems

Prove that the general equation of the seventh degree cannot be solved using functions of only two variables....

If $\alpha$ is algebraic and irrational, and $\beta$ is algebraic and irrational, is $\alpha^\beta$ transcendental?...

Extend the theory of quadratic forms with algebraic numerical coefficients....

Is the ring of invariants of a linear algebraic group acting on a polynomial ring always finitely generated?...

Rigorously justify Schubert's enumerative geometry....