Journal of the
Korean Mathematical Society JKMS

Let $m, n$ be positive integers or monic polynomials in $\mathbb F_{q}[T]$ with $n | m$. Let $K_m$ and $K_m^+$ be the $m$-th cyclotomic field and its maximal real subfield, respectively. In this paper we define two matrices $\mathcal D_{m,n}^+$ and $\mathcal D_{m,n}^-$ whose determinants give us the ratios $\frac{h(\mathcal O_{K_m^+})}{h(\mathcal O_{K_n^+})}$ and $\frac{h^{-}(\mathcal O_{K_m})}{h^{-}(\mathcal O_{K_n})}$ with some factors, respectively.

Keywords: cyclotomic unit, cyclotomic function field

MSC numbers: 11R18, 11R60