John Baez and James Dolan, 2023-03-23

Studying abelian extensions of fields using torsors. The case of quadratic extensions of the rationals was relatively easy because ℤ/2 'splits' over the rational numbers: that is, all irreducible representations of ℤ/2 on rational vector spaces are 1-dimensional. To extend the program to ℤ/3, which does not split over the rationals, we conduct some preliminary investigations of subfields of the 63rd cyclotomic field, which correspond to subgroups of GL(1, ℤ/63) = ℤ/6 × ℤ/6. There are 30 such subgroups, and we construct a chart of 12 of them mimicking the method used on February 16th, 2023 for the 20th cyclotomic field:
• John Baez and James Do...
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