A Drinfeld module over an 
-field 
is a ring homomorphism
of 
-algebras 
satisfying the constant-term and nonscalar
conditions below. Here, 
is a finite field with 
elements, and 
is the coordinate ring
of 
for a smooth projective geometrically integral
algebraic curve 
over 
and a chosen closed point 
. An 
-field is a field 
equipped with a ring homomorphism 
. The ring 
consists of 
-linear additive polynomials,
with multiplication given by composition
and 
.
For every 
,
the constant coefficient of 
as a polynomial in 
must be 
, and 
must have positive 
-degree for at least
one 
.
Its rank 
is the positive integer satisfying 
for every nonzero 
, where 
is the number of elements
of 
.
In particular, when ![A=F_q[T]](/images/equations/DrinfeldModule/Inline30.svg)
,
the rank is the 
-degree of 
. Every Drinfeld module is of dimension
1 regardless of its rank. Rank 2 gives the analogy with an elliptic
curve (Poonen 2021, §§2.2 and 3.1-3.4).
