A bijection is an isomorphism in the category of sets, meaning that it is just an invertible one-to-one mapping between two sets.Many categories can be thought of as sets with additional structure (e.g. groups, rings, differentiable functions etc.) and an isomorphism requires the structure of the category to be preserved. For example if you are mapping in the category of groups, then you need to map the identity element of the first group to the identity of the second group and you need to preserve the group law so that f(a*b) = f(a).f(b) where * is the group operation of the first group and . is the operation in the second group.

The fact that there is a hint in the instructions to look at the wikipedia page for a bijection rather than the category theory definition of an isomorphism is actually a fairly subtle clue how to solve one of the parts of the kata which can only be solved by possibly breaking the isomorphism structure.

OK - figured out the problem.

Although the show representation was identical, my internal representation of polynomials was ambiguous.
To fix this I needed to implement Eq.

In the field multiplication test I got the result:
Expected x^6 + x^5 + x^4 + x^3 + x^1 + x^0 but got x^6 + x^5 + x^4 + x^3 + x^1 + x^0

I can't see any difference here. What's the problem?