  We introduce the concept of subalgebra spectrum, $Sp(A)$, for a subalgebra
$A$ of finite codimension in $\mathbb{K}[x]$. The spectrum is a subset of the
underlying field. We also introduce a tool, the characteristic polynomial of
$A$, which has the spectrum as its set of zeroes. The characteristic polynomial
can be computed from the generators of $A$, thus allowing us to find the
spectrum of an algebra given by generators. We proceed by using the spectrum to
get descriptions of subalgebras of finite codimension. More precisely we show
that $A$ can be described by a set of conditions that each is either of the
type $f(\alpha)=f(\beta)$ for $\alpha,\beta$ in $Sp(A)$ or of the type stating
that some sum of derivatives of different orders evaluated in elements of
$Sp(A)$ equals zero. We use this type of conditions to, by an inductive
process, find explicit descriptions of subalgebras of codimension up to three.
These descriptions also include SAGBI bases for each family of subalgebras.
