# Algebraic function

Algebraic functions are a special class of functions that are examined in particular in the mathematical sub-area of algebra . They are the solution to an algebraic equation . Functions that are not algebraic are called transcendent functions .

The theory of algebraic functions was developed in the past from the three mathematical sub-areas of function theory , arithmetic algebraic geometry and algebraic geometry .

## definition

A function in variables is called an algebraic function if there is an irreducible polynomial in variables and coefficients in a field such that the algebraic equation${\ displaystyle y = f (x_ {1}, \ dotsc, x_ {n})}$${\ displaystyle n}$ ${\ displaystyle P}$${\ displaystyle n + 1}$${\ displaystyle f}$

${\ displaystyle P (f (x_ {1}, \ dotsc, x_ {n}), x_ {1}, \ dotsc, x_ {n}) = 0}$

solves.

So a function of a variable is algebraic if it has the equation ${\ displaystyle y = f (x)}$

${\ displaystyle P_ {n} (x) y ^ {n} + \ dotsb + P_ {1} (x) y + P_ {0} (x) = 0}$

fulfilled, where polynomials are in the variable . ${\ displaystyle P_ {1}, \ dotsc, P_ {n}}$${\ displaystyle x}$

## properties

• Since the definition required that the polynomials are irreducible, it can be proven that for every algebraic function there is exactly one irreducible polynomial with the exception of one constant . The degree of the polynomial in the variable is then called the degree of the algebraic function.${\ displaystyle y = f (x_ {1}, \ dotsc, x_ {n})}$${\ displaystyle P}$${\ displaystyle P (y, x_ {1}, \ dotsc, x_ {n}) = 0}$${\ displaystyle P}$${\ displaystyle y}$
• For degrees , all algebraic functions can be represented as rational functions and for degrees , and they can all be represented as the square or cube roots of a rational function. In general, this is not possible for grades .${\ displaystyle 1}$${\ displaystyle 2}$${\ displaystyle 3}$${\ displaystyle 4}$${\ displaystyle k> 4}$
• Algebraic functions of a variable over the field of complex numbers are meromorphic .${\ displaystyle \ mathbb {C}}$

## Transcendent functions

A function is called transcendent if it is not algebraic. These include, for example