I do not know whether it is logic or number theory, but Hilbert's 10th problem over rationals ("is there an algorithm to decide whether a given polynomial equation $f(x_1,\dots,x_n)$ with rational coefficients have a rational solution?") is still open, on the contrast with negative solution to equations over integers (DPRM theorem.)

I do not know whether it is logic or number theory, but Hilbert's 10th problem over rationals ("is there an algorithm to decide whether a given polynomial equation $f(x_1,\dots,x_n)=0$ with rational coefficients have a rational solution?") is still open, on the contrast with negative solution to equations over integers (DPRM theorem.)