The anti-field-descent method

The essential fact behind the so-called field-descent method is that certain cyclotomic integers necessarily are contained in relatively small fields and thus must have relatively small complex modulus. In this paper, we develop a method which reveals a complementary phenomenon: certain cyclotomic integers cannot be contained in relatively small fields and thus must have relatively large complex modulus.This method, in particular, yields progress towards the circulant Hadamard matrix conjecture. In fact, we show that such matrices give rise to certain “twisted cyclotomic integers” which often have small complex modulus, but are not contained in small fields. Hence our “anti-field-descent” method provides new necessary conditions for the existence of circulant Hadamard matrices. The application of the new conditions to previously open cases of Barker sequences shows that there is no Barker sequence of length ℓ   with 13<ℓ≤4⋅103313<ℓ≤4⋅1033. Furthermore, 229,682 of the 237,807 known open cases of the Barker sequence conjecture are ruled out.