fields - roots of polynomials with multiple-of-unity coefficients implies algebraically closed?

Suppose a field has roots of all polynomials whose coefficients are 0, 0+1, 0+1+1, 0+1+1+1, 0+1+1+1+1, etc or additive inverses thereof. Is such a field necessarily algebraically closed?

The algebraic numbers are produced by starting with the rationals and closing under roots of polynomials as described above (ie integer coefficients). The algebraic numbers are algebraically closed -- any polynomial with any coefficients in the algebraic numbers has a root (not just polynomials with integer coefficients). Is this phenomenon specific to the algebraic numbers, or is it true for fields in general?

Thanks,