The complex numbers are a degree 2 field extension over the real numbers.
The general theorem is that for a field k, and an irreducible (meaning it can't be factored with coefficients in k) polynomial p(x) with coefficients in k, the smallest field containing a root of p(x) and k is a vector space (over k) of dimension deg p(x). The irreducible polynomial corresponding to i is x^2 + 1 = 0.
Similarly, a finite field of order q = p^r can be constructed with an irreducible polynomial of degree r with coefficients in the prime field of order p.
The general theorem is that for a field k, and an irreducible (meaning it can't be factored with coefficients in k) polynomial p(x) with coefficients in k, the smallest field containing a root of p(x) and k is a vector space (over k) of dimension deg p(x). The irreducible polynomial corresponding to i is x^2 + 1 = 0.
Similarly, a finite field of order q = p^r can be constructed with an irreducible polynomial of degree r with coefficients in the prime field of order p.