Group Isomorphism from $\mathbb{Q}[X]^2$ to $\mathbb{Q}[X]$.

Question

I'm trying to find a bijective group homomorphism from $\mathbb{Q}[X]^2$ to $\mathbb{Q}[X]$, where $\mathbb{Q}[X]$ is the group of polynomials with rational coefficients whose operation is given by addition of polynomials. So far, I've tried different examples of functions $\Phi: \mathbb{Q}[X]^2 \rightarrow \mathbb{Q}[X]$, and the only homomorphisms I have found are of the form $\Phi(x, y) = px + qy$, where $p, q \in \mathbb{Q}[X]$ are fixed. However, these are not injective, since $\Phi(-q, p) = 0$; therefore, these are not bijective. Can anyone give some ideas of other ways to construct a homomorphism which would potentially be bijective? Thanks in advance for any help.

Answer 1

Since you’re only considering the additive structure of these polynomials, not their multiplicative structure, they’re really nothing more than finite sequences of rational numbers. Thus, all you need is some way to interleave two such sequences. One such way has been suggested in the comments, alternatingly taking elements from the two sequences, but you could use any other scheme, e.g. alternatingly taking two elements from each sequence, alternatingly taking one element from one sequence and two from the other, etc.