A note on a refined formula of the relative class number of an imaginary abelian field

Key Points

• This work aims to refine the formula for the relative class number of imaginary abelian fields using Wiles' theorem.
• Referenced an existing formula for an imaginary abelian field from prior research.
• Applied Wiles' theorem to a specific abelian CM-extension of a totally real algebraic number field.
• Analyzed the implications of the total real nature of the maximal abelian p-extension contained in the field.
• The refined formula aligns with known results when the Iwabawa polynomials have no non-trivial zeros.
• Demonstrated a new relationship between abelian p-extensions and class numbers in imaginary abelian fields.