An algebraic function of 
variables over a field 
is a function 
for which there is a nonzero polynomial 
with coefficients
in 
such that
 |
When 
,
clearing denominators shows that 
may equivalently be taken to have coefficients
in the integers. Such a polynomial
is called a defining polynomial of 
over 
.
A defining polynomial need not determine a unique algebraic function. For example, 
defines both 
and 
on 
. A particular branch can
be selected by an additional condition, such as 
for the principal
square root, or by exact root isolation data.
For one complex variable, let 
be a polynomial with complex coefficients. If 
and 
, the implicit
function theorem gives a unique local analytic branch 
through 
. Implicit
differentiation gives
 |
where 
and 
denote partial derivatives. This local branch
need not agree throughout its domain with a fixed ordering
of the roots, and analytic
continuation can permute branches around a branch
point.
Every expression obtained from polynomials with coefficients in 
by finitely many additions, subtractions, multiplications,
divisions, and rational powers
is algebraic over 
.
A defining polynomial for such an expression with
radicals can be constructed recursively using resultants
(Zippel 1993, Maaz and Strzeboński 2025). Not every algebraic function can be
expressed using radicals, as follows from Abel's
impossibility theorem. A function which is not algebraic over 
is called a transcendental
function over 
.
