  We study the degree of irreducible morphisms in any Auslander-Reiten
component of a finite dimensional algebra over an algebraically closed field.
We give a characterization for an irreducible morphism to have finite left (or
right) degree. This is used to prove our main theorem: An algebra is of finite
representation type if and only if for every indecomposable projective the
inclusion of the radical in the projective has finite right degree, which is
equivalent to require that for every indecomposable injective the epimorphism
from the injective to its quotient by its socle has finite left degree. We also
apply the techniques that we develop: We study when the non-zero composite of a
path of $n$ irreducible morphisms between indecomposable modules lies in the
$n+1$-th power of the radical; and we study the same problem for sums of such
paths when they are sectional, thus proving a generalisation of a pioneer
result of Igusa and Todorov on the composite of a sectional path.
