Narrow Results

Let $q$ be a prime power and $f\in {𝔽}_q[x]$. In this paper we complete the classification of good polynomials of degree $6$ that achieve the best possible asymptotics (with an explicit error term) for the number of totally split places. Moreover, for degrees up to $6$, we provide an explicit lower bound and an asymptotic estimate for the number of totally split places of $f$. Finally, we prove the general fact that the number $a_i$ of $t_0\in {𝔽}_{q^i}$ for which $f-t_0$ splits obeys a linear recurring sequence. For any $f\in {𝔽}_q[x]$, this allows for the computation of $a_i$ for large $i$ by only computing a recurrence sequence over $\mathbb{Z}$.

The aim of this paper is to present some connections of infinite Cogalois Theory with Clifford extensions and Hopf algebras.

A formula for computing the discriminant of any number field K ⊂ ℚ(ζ_2^r), with r ≥ 3, is derived. The formula consists of two expressions, depending on whether K is cyclotomic or not. However, both expressions depend solely on m, the degree of K over ℚ, and they are derived from the Conductor-Discriminant Formula for Abelian extensions of ℚ.

We present a theory for splitting algebras of monic polynomials over rings, and apply the results to symmetric functions, and Galois theory.

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.

For a proper subfield K of we show the existence of an algebraic number α such that no power αⁿ, n ≥ 1, lies in K. As an application it is shown that these numbers, multiplied by convenient Gaussian numbers, can be written in the form P(T)^Q(T) for some transcendental numbers T where P and Q are arbitrarily prescribed nonconstant rational functions over .

A Hopf Galois structure on a finite field extension $L/K$ is a pair $(H,\mu )$, where $H$ is a finite cocommutative $K$-Hopf algebra and $\mu$ a Hopf action. In this paper, we present a program written in the computational algebra system Magma which gives all Hopf Galois structures on separable field extensions of a given degree and several properties of those. We show a table which summarizes the program results. Besides, for separable field extensions of degree $2p^n$, with $p$ an odd prime number, we prove that the occurrence of some type of Hopf Galois structure may either imply or exclude the occurrence of some other type. In particular, for separable field extensions of degree $2p^2$, we determine exactly the possible sets of Hopf Galois structure types.

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra $R^{\underline{coA}}$ with a certain subalgebra of the smash product $R\#Â$. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.