[math-fun] Reciprocals in Q(2^(1/3))

Suppose w is the real cube root of 2, w^3 = 2. Then for a0, a1, a2 rational, numbers of the form a0 + a1*w + a1*w^2 form a field. What I want is the formula for the multiplicative inverse in this field. It's elementary algebra, but it is voluminous and I keep making errors when I try to do it by hand. Any computer algebra system could probably instantly solve for b0, b1, b2 subject to (a0 + a1w + a2ww)(b0 + b1w + b2ww) = 1. Can somebody do me a favor and give me formulas for the b's in terms of the a's? Thank you!