Z is not a field: most of its elements are not invertible.

On the other hand, Q, R, C are fields.

Z/nZ is a field if and only if n is a prime number. See Section 2.1.

Z[X], Q[X], R[X], C[X] are not fields: X does not have an inverse.

If R = Q, R, C, or Z/pZ for some prime p, and f is a polynomial in R[X], then R[X]/(f) is a field if and only f is irreducible in R[X]. See Section 4.5.

The ring R = Z + Zi of Gaussian integers is not a field.

For instance, the element 1 + i has no inverse: if a + bi were its inverse, then

2a + 2bi = (a + bi) 2 = (a + bi) (1 + i) (1 - i) = 1 - i,

whence 2a = 1, which contradicts a Z.

The variation Q + Qi however, is a field. Can you find the inverse of an arbitrary nonzero element?