Narrow Results

We give a construction of the genus field for Kummer ${\ell }^n$-cyclic extensions of rational congruence function fields, where $\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $\ell$-cyclic extension. Finally, we study the extension ${(K_1{K}_2)}_{𝔤𝔢}/{(K_1)}_{𝔤𝔢}{(K_2)}_{𝔤𝔢}$, for $K_1$, $K_2$ abelian extensions of $k$.

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let , where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field k_p((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

In this work, $p$-dimensional point-lattices are constructed from a number field $L$, where $L/\mathbb{Q}$ is a cyclic extension of degree $p$, an odd unramified prime in $L/\mathbb{Q}$. Families of lattices whose packing densities asymptotically approach those of ${\mathrm{\Lambda }}_p$, $p=3,5,7$, are explicitly exhibited.

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.