  For a coalgebra $C_k$ over field $k$, we define the "coalgebra extension
problem" as the question: what multiplication laws can we define on $C_k$ to
make it a bialgebra over $k$? This paper answers this existence-uniqueness
question for certain coalgebras called "circulant coalgebras". We begin with
the trigonometric coalgebra, comparing and contrasting with the
group-(bi)algebra $k[\mathbb Z/2]$. This leads to a generalization, the dual
coalgebra to the group-algebra $k[\mathbb Z/p]$, which we then investigate. We
show connections with other questions, motivating us to answer to the coalgebra
extension problem for these families. The answer depends interestingly on the
base field $k$'s characteristic.
  Along similar lines, we investigate the algebraic group $S^1$ over arbitrary
$k$. We find that similar complications arise in characteristic 2. We explore
this, motivated (by quantum groups) by the question of whether or not
$\mathcal{O}(S^1)$ is pointed. We give a very explicit conjecture in terms of
the Chebyshev polynomials of trigonometry. We end by constructing a formal
group object, in a certain monoidal category of modules of $k[[h]]$, as a
$2^{\text{nd}}$ order deformation of $k[t]$.
