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In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a 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,...,xn) 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 Y 2 − X 3 and form the field of fractions of the quotient ring k [X,Y]/(Y 2 − X 3).
We show that the transcendence degree of a real function field over an arbitrary real base field is a strict lower bound for its Pythagoras number and a weak lower bound for all its higher Pythagoras
A recently found local-global principle for quadratic forms over function fields of curves over a complete discretely valued field is applied to the study of quadratic forms, sums of squares, and rela
Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. A