Algebraic function field

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. Example 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). This is a function field of one variable over k; it can also be written as k(X)(\sqrt{X^3}) (with degree 2 over k(X)) or as k(Y)(\sqrt[3]{Y^2}) (with degree 3 over k(Y)). We see that the degree of an algebraic function field is not a well-defined notion. Category structure The algebraic funct

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Sums Of Squares In Algebraic Function Fields Over A Complete Discretely Valued Field

David Maximilian Grimm

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 related field invariants.

Pacific Journal Mathematics2014

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. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.

Springer-Verlag2013

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 numbers.

European Mathematical Soc2015

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