Determining irreducibility and ramification groups for an additive extension of the rational function field.
Let \(K\) be a subfield of \(\overline{\mathbb F}_p\), not necessarily proper. If \(V\) is a finite subgroup of the additive group of \(K\), let \(a_V(T)\) denote the additive polynomial \(\prod_{v\in V} (T-v)\). The author considers the extension of \(K(x)\) obtained by adjoining a root of \(a_V(T)-f(x)\), where \(f(x)\in K[x]\). First, the following criterion for the irreducibility of \(a_V(T)-f(x)\) is given. A term in \(f(x)\) is called a coprime term if its degree is coprime to \(p\), and a coprime term of degree \(d\) is called totally coprime if \(f(x)\) has no terms of degree \(dp^i\) for \(i>0\). The coprime degree of \(f(x)\) is the degree of the largest totally coprime term if \(f(x)\) has any totally coprime terms, and is zero otherwise. The author proves that if the coprime degree of \(f(x)\) is positive, then the polynomial \(a_V(T)-f(x)\) is irreducible over \(K(x)\) for every finite subgroup \(V\) of the additive group of \(K\). In many cases, this irreducibility criterion is simpler to apply than the Artin-Schreier Theorem. If the coprime degree of \(f\) is greater than \({\deg}(f)/p\), then the author determines the sequence of ramification groups at the place at infinity and gives the genus of the above extension of \(K(x)\). This theory is applied when \(K={\mathbb F}_{q^n}, a_V(T)=\sum_{0\leq j\leq n-1}T^{q^j}\), and \(f(x)\) is an \((n,q)\)-elementary symmetric polynomial, a type of polynomial that also has been considered by \textit{A. Garcia} and \textit{H. Stichtenoth} [Finite Fields Appl. 5, 424--435 (1999; Zbl 0965.11027)]. The author computes the number of rational places of the extension in this situation, and it is shown that the extension has relatively many rational places when \(f(x)=\sum x^{q^j+q^k}\), where \(j<k\) and \(0\leq j,k\leq n-1\). An appendix contains more details on \((n,q)\)-symmetric polynomials.
function fields with many rational places
scientific article; zbMATH DE number 1891442