Systems of forms in C_i fields

Abstract

The notion of algebraically closed fields can be generalised by C_i fields. These are fields with the following property. Let f be a homogenous polynomial with coefficients in a field F and i a natural number. Then f has a non trivial zero in F if the number of variables of f is greater than the i-th power of the degree of f. A field is C_0 if and only if it is algebraically closed. The definition of a C_i field can thus be seen as a measurement of how close a field is to being algebraically closed. One can generalise the notion of C_i fields to strongly C_i fields by considering polynomials without constant term instead of homogenous polynomials. In this thesis the algebraic properties of (strongly) C_i fields are investigated. First, it is proven that a field is C_0 if and only if it is algebraically closed. Then it is shown that finite fields are strongly C_1. We investigate under what circumstances the notion of C_i fields can be extended to non trivial common zeros of systems of homogenous polynomials. These results are used to show that algebraic extensions of C_i fields are also C_i. An important open question about C_i fields is whether C_i fields are also strongly C_i. We show that this is the case for C_0 fields and provide a condition under which a C_i field is strongly C_(i+1).