Tsen rank

Last updated April 26, 2023

In mathematics, the Tsen rank of a field describes conditions under which a system of polynomial equations must have a solution in the field. The concept is named for C. C. Tsen, who introduced their study in 1936.

Properties

A field has Tsen rank zero if and only if it is algebraically closed.
A finite field has Tsen rank 1: this is the Chevalley–Warning theorem.
If F is algebraically closed then rational function field F(X) has Tsen rank 1.
If F has Tsen rank i, then the rational function field F(X) has Tsen rank at most i + 1.
If F has Tsen rank i, then an algebraic extension of F has Tsen rank at most i.
If F has Tsen rank i, then an extension of F of transcendence degree k has Tsen rank at most i + k.
There exist fields of Tsen rank i for every integer i ≥ 0.

Norm form

We define a norm form of level i on a field F to be a homogeneous polynomial of degree d in n=dⁱ variables with only the trivial zero over F (we exclude the case n=d=1). The existence of a norm form on level i on F implies that F is of Tsen rank at least i − 1. If E is an extension of F of finite degree n > 1, then the field norm form for E/F is a norm form of level 1. If F admits a norm form of level i then the rational function field F(X) admits a norm form of level i + 1. This allows us to demonstrate the existence of fields of any given Tsen rank.

Diophantine dimension

The Diophantine dimension of a field is the smallest natural number k, if it exists, such that the field of is class C_k: that is, such that any homogeneous polynomial of degree d in N variables has a non-trivial zero whenever N > d^k. Algebraically closed fields are of Diophantine dimension 0; quasi-algebraically closed fields of dimension 1.^[1]

Clearly if a field is T_i then it is C_i, and T₀ and C₀ are equivalent, each being equivalent to being algebraically closed. It is not known whether Tsen rank and Diophantine dimension are equal in general.

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References

↑ Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.

Tsen, C. (1936). "Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper". J. Chinese Math. Soc. 171: 81–92. Zbl 0015.38803.
Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4.

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[NSW361-1] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2008). Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften. Vol. 323 (2nd ed.). Springer-Verlag. p. 361. ISBN 978-3-540-37888-4.

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