Algebraic function field

In mathematics, an (algebraic) function field of n variables over the field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K=k(x₁,...,x_n) of rational functions in n variables over k.

As an example, in the polynomial ring k[X,Y] consider the ideal generated by the irreducible polynomial Y²−X³ and form the field of fractions of the quotient ring k[X,Y]/(Y²−X³). This is a function field of one variable over k; it can also be written as ${\displaystyle k(X)({\sqrt {X^{3}}})}$ (with degree 2 over ${\displaystyle k(X)}$) or as ${\displaystyle k(Y)({\sqrt[{3}]{Y^{2}}})}$ (with degree 3 over ${\displaystyle k(Y)}$). We see that the degree of an algebraic function field is not a well-defined notion.

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