TITLE: Polynomial group Laws on $\mathbb{R}^2$
QUESTION [5 upvotes]: When students are first learning about groups, a classic example of a group that is not defined as a set of functions is the group whose underlying set is $\mathbb{R}\setminus-1$, and whose operation is $x*y=x+y+xy$. This naturally leads one to wonder about what other polynomials in two variables give rise to a group law on $\mathbb{R}$. Is there any nice criteria for such polynomials, or, in the case that there is not, are there any nice classes of polynomials that are group laws?

REPLY [3 votes]: All the above observations and much much more can be found in the classic paper:
Hinrichs, L.; Niven, I.; Eynden, C.L.V., Fields defined by polynomials, Pac. J. Math. 14, 537-545 (1964). ZBL0122.29103.
Here the authors prove that $f(x,y)$ is a polynomially defined group law on an infinite field $k$ if and only if $f(x,y) = x+y+c$ for some constant $c$ in $k$. They go on to characterize polynomially defined field operations on $R\times R$ as well.
Very useful paper from the point of view of this topic especially at a beginning level.
- R. Padmanabhan, University of Manitoba.