arXiv subjects

Tara L. Smith

Publications and source records attributed to Tara L. Smith.

2 recordsLinked to original sources

arXiv2001

Additive structure of multiplicative subgroups of fields and Galois theory

One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro-2-Galois groups of nilpotency class at most 2, and of exponent at most 4. This work extends some powerful methods and techniques from formally real fields to general fields of characteristic not 2.

math.GR↗

arXiv2000

Galois Groups Over Nonrigid Fields

Let $F$ be a field with characteristic $\neq 2$. We show that $F$ is a nonrigid field if and only if certain small 2-groups occur as Galois groups over $F$. These results provide new "automatic realizability" results for Galois groups over $F$. The groups we consider demonstrate the inequality of two particular metabelian 2-extensions of $F$ which are unequal precisely when $F$ is a nonrigid field. Using known results on connections between rigidity and existence of certain valuations, we obtain Galois-theoretic criteria for the existence of these valuations.

math.NT↗