Abstract
AbstractWe discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .
Highlights
In a 1939 Crelle paper [16], the Hungarian mathematician László Rédei introduced a trilinear quadratic symbol [a, b, c] ∈ {±1} for quadratic discriminants a, b ∈ Z and positive squarefree integers c satisfying a number of conditions
Linear algebra gives the 8-rank of C in terms of a matrix R8 = R8(D) over F2 (Theorem 4·1), but this time its entries are (F2-valued) Rédei symbols [d1, d2, m], given in Definition 4·4 as the Artin symbo√l of an ambiguous ideal in √K of √norm m in an unramified cyclic quartic extension K = Q( d1d2) ⊂ Fd1,d2 having Q( d1, d2) as its intermediate
A product m of distinct ramified primes of K is characterized by the squarefree divisor m|D arising as its norm√, and√the residue class of a quartic character ψ in C[4]/C[2] by the invariant field E = Q( d1, d2) of the quadratic character 2ψ corresponding to a decomposition D = d1d2 ‘of the second kind’. This leads to a classical notation for the entries ψ([m]) in (23) as Rédei symbols
Summary
In a 1939 Crelle paper [16], the Hungarian mathematician László Rédei introduced a trilinear quadratic symbol [a, b, c] ∈ {±1} for quadratic discriminants a, b ∈ Z and positive squarefree integers c satisfying a number of conditions. Linear algebra gives the 8-rank of C in terms of a matrix R8 = R8(D) over F2 (Theorem 4·1), but this time its entries are (F2-valued) Rédei symbols [d1, d2, m], given in Definition 4·4 as the Artin symbo√l of an ambiguous ideal in √K of √norm m in an unramified cyclic quartic extension K = Q( d1d2) ⊂ Fd1,d2 having Q( d1, d2) as its intermediate.
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