This isn't essential, but it's worth noting that the construction of Galois fields is basically done in the same way as a more familiar one: building the complex numbers from the real numbers. In that case, the new "numbers" are defined to be polynomials with real coefficients, with addition and multiplication being performed modulo the polynomial x^2+1. This has the effect of equating x^2+1 with 0, since the division (x^2+1)/(x^2+1) has remainder 0. With this rule, x is now a square root of -1; of course we usually write i instead of x. In terms of the compact notation mentioned in the article, the complex numbers are the same thing as R[x]/(x^2+1).
The Galois field case can be thought of in the same way, as long as a little care is taken with the choice of polynomial. When the coefficients come from GF(2), there's not much point in using the polynomial x^2+1 as above, because x^2+1 = x^2+2x+1 = (x+1)^2. Forcing x^2+1 = (x+1)^2 to be 0 would basically just have the effect of setting x = -1 = 1, so we don't get any new numbers. [Technically, 0, 1, x, 1+x would still be distinct in this construction, but it doesn't result in a field since 1+x would have no multiplicative inverse.] As explained in the article, the polynomial should be irreducible to avoid this problem, so x^2+x+1 works to build GF(4) from GF(2). But this is the only difference from complex numbers: we can think of GF(4) as being GF(2) with an added "fictional number" h satisfying h^2+h+1 = 0 (i.e. h^2 = h+1). The elements of GF(4) are therefore numbers ah+b where a,b are in GF(2), multiplied just like complex numbers except that we simplify using the rule h^2 = h+1 instead of i^2 = -1.
In the Galois field case, lots of different polynomials appear because (1) we need a degree k irreducible polynomial to construct GF(p^k) from GF(p) and (2) there's not really an obvious "simplest" such polynomial to use, unlike in the case of the complex numbers C. In that case, a miraculous fact intervenes to save us from a similar zoo of polynomials: as soon as we add the one "fictional number" i, all polynomials with complex coefficients have roots in terms of it, so there are no more fictional numbers to be created this way starting from C.
The Galois field case can be thought of in the same way, as long as a little care is taken with the choice of polynomial. When the coefficients come from GF(2), there's not much point in using the polynomial x^2+1 as above, because x^2+1 = x^2+2x+1 = (x+1)^2. Forcing x^2+1 = (x+1)^2 to be 0 would basically just have the effect of setting x = -1 = 1, so we don't get any new numbers. [Technically, 0, 1, x, 1+x would still be distinct in this construction, but it doesn't result in a field since 1+x would have no multiplicative inverse.] As explained in the article, the polynomial should be irreducible to avoid this problem, so x^2+x+1 works to build GF(4) from GF(2). But this is the only difference from complex numbers: we can think of GF(4) as being GF(2) with an added "fictional number" h satisfying h^2+h+1 = 0 (i.e. h^2 = h+1). The elements of GF(4) are therefore numbers ah+b where a,b are in GF(2), multiplied just like complex numbers except that we simplify using the rule h^2 = h+1 instead of i^2 = -1.
In the Galois field case, lots of different polynomials appear because (1) we need a degree k irreducible polynomial to construct GF(p^k) from GF(p) and (2) there's not really an obvious "simplest" such polynomial to use, unlike in the case of the complex numbers C. In that case, a miraculous fact intervenes to save us from a similar zoo of polynomials: as soon as we add the one "fictional number" i, all polynomials with complex coefficients have roots in terms of it, so there are no more fictional numbers to be created this way starting from C.