Small generators of function fields

Martin Widmer

Displaying similar documents to “Small generators of function fields”

Computation of 2-groups of positive classes of exceptional number fields

Jean-François Jaulent, Sebastian Pauli, Michael E. Pohst, Florence Soriano–Gafiuk (2008)

Journal de Théorie des Nombres de Bordeaux

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We present an algorithm for computing the 2-group ${𝒞 ℓ}_{F}^{p o s}$ of the positive divisor classes in case the number field $F$ has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel $W K_{2} (F)$ in $K_{2} (F)$ .

Thomas’ conjecture over function fields

Volker Ziegler (2007)

Journal de Théorie des Nombres de Bordeaux

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Thomas’ conjecture is, given monic polynomials $p_{1},$ $..., p_{d} \in ℤ [a]$ with $0 < deg p_{1} < \dots < deg p_{d}$ , then the Thue equation (over the rational integers) $(X - p_{1} (a) Y) \dots (X - p_{d} (a) Y) + Y^{d} = 1$ has only trivial solutions, provided $a \geq a_{0}$ with effective computable $a_{0}$ . We consider a function field analogue of Thomas’ conjecture in case of degree $d = 3$ . Moreover we find a counterexample to Thomas’ conjecture for $d = 3$ .

Explicit construction of integral bases of radical function fields

Qingquan Wu (2010)

Journal de Théorie des Nombres de Bordeaux

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We give an explicit construction of an integral basis for a radical function field $K = k (t, ρ)$ , where $ρ^{n} = D \in k [t]$ , under the assumptions $[K : k (t)] = n$ and $c h a r (k) ∤ n$ . The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$ -signatures of a radical function field are also discussed in this paper.

Approximation exponents for algebraic functions in positive characteristic

Bernard de Mathan (1992)

Acta Arithmetica

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In this paper, we study rational approximations for algebraic functions in characteristic p > 0. We obtain results for elements satisfying an equation of the type $α = (A α^{q} + B) / (C α^{q} + D)$ , where q is a power of p.

Formulae for the singularities at infinity of plane algebraic curves.

Gwoździewicz, Janusz, Płoski, Arkadiusz (2001)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

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Constructing class fields over local fields

Sebastian Pauli (2006)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a $𝔭$ -adic field. We give an explicit characterization of the abelian extensions of $K$ of degree $p$ by relating the coefficients of the generating polynomials of extensions $L / K$ of degree $p$ to the exponents of generators of the norm group $N_{L / K} (L^{*})$ . This is applied in an algorithm for the construction of class fields of degree $p^{m}$ , which yields an algorithm for the computation of class fields in general.