# Redei symbols and governing fields

#### Abstract

In 1939, L. Rédei [20] introduced a triple symbol [*a*_{ 1}, *a*_{2}, *a*_{3}] for integers *a*_{1}, *a*_{2} and * a*_{3}, to investigate the decomposition of primes in certain dihedral extensions of Q . In this thesis we give an alternative, and more general, definition of these symbols. One of the differences between the two approaches is that we use ideles instead of ideals, which allows us to take the infinite primes into account. This leads to nicer properties of the symbol; in fact, when the symbol is "pure'', then we show that it is invariant under all permutations of its entries and linear in all terms. ^ We next study the 8-rank of the narrow class group *Cl*^{ +} (*F*) for a quadratic number field *F,* that is, the F2 -dimension of *Cl*^{+} (*F*)^{ 4}/*Cl*^{+} (*F*)^{ 8}, and show that it is determined by certain Rédei symbols. ^ For any fixed squarefree integer *d* and *n* ≥ 2, Cohn and Lagarias [4] conjectured the existence of a field Ω_{ 2n} (*d*) such that the 2^{n}-rank of *Cl*^{+} (*F*) for *F * = Q ( dp ) depends only on the Frobenius symbol of *p* in Ω_{ 2n} (*d*)/ Q . We show that the properties of the Rédei symbol imply the conjecture of Cohn and Lagarias for *n* = 3. Here the difference in the definitions of Rédei symbols turns out to be essential. ^ We also study the relationship between Rédei symbols and triple Massey products, extending results by Morishita [10, 11] and Vogel [25], who considered only the special case when the entries in the Rédei symbols are primes *p* with *p* ≡1 mod 4. ^

#### Subject Area

Mathematics

#### Recommended Citation

Jens Corsman,
"Redei symbols and governing fields"
(January 1, 2007).
*ETD Collection for McMaster University.*
Paper AAINR36051.

http://digitalcommons.mcmaster.ca/dissertations/AAINR36051