Structure Predicates

IsField(R): FldFun -> BoolElt

IsEuclideanDomain(R): FldFun -> BoolElt

IsField(O): RngFunOrd -> BoolElt

IsPID(R): FldFun -> BoolElt

IsUFD(R): FldFun -> BoolElt

IsDivisionRing(R): FldFun -> BoolElt

IsEuclideanRing(R): FldFun -> BoolElt

IsDivisionRing(O): RngFunOrd -> BoolElt

IsPrincipalIdealRing(R): FldFun -> BoolElt

IsDomain(R): FldFun -> BoolElt

IsDomain(O): RngFunOrd -> BoolElt

F eq G: FldFunG, FldFunG -> BoolElt

F ne G: FldFunG, FldFunG -> BoolElt

O1 eq O2: RngFunOrd, Rng -> BoolElt

O1 ne O2: RngFunOrd, Rng -> BoolElt

O1 subset O2: RngFunOrd, RngFunOrd -> BoolElt

Return whether \(O1\) is a subset of \(O2\).

IsGlobal(F): FldFunG -> BoolElt

Returns true if and only if the algebraic function field \(F/k\) is global, i.e. the constant field is a finite field; false otherwise.

IsRationalFunctionField(F): FldFunG -> BoolElt

Return true if the function field \(F\) is isomorphic to a rational function field, (i.e. \(F\) is only trivially algebraic).

IsFiniteOrder(O): RngFunOrd -> BoolElt

Given an order \(O\) of a function field, return true if and only if the bottom coefficient ring of \(O\) is a polynomial ring.

IsEquationOrder(O): RngFunOrd -> BoolElt

Given an order \(O\) of a function field, return true if and only if the order \(O\) is an equation order (i.e. it has been defined by a polynomial and so has a power basis).

IsAbsoluteOrder(O): RngFunOrd -> BoolElt

Return false if the order \(O\) is an extension of another order, otherwise true.

IsMaximal(O): RngFunOrd -> BoolElt

Given an order \(O\) of a function field, return true if and only if the order \(O\) is maximal in its field of fractions.

IsTamelyRamified(O): RngFunOrd -> BoolElt

Return whether the order \(O\) is tamely ramified, i.e. no prime ideal of \(O\) has residue field with characteristic dividing its ramification index.

IsTotallyRamified(O): RngFunOrd -> BoolElt

Return whether there is an ideal of the order \(O\) which is totally ramified, i.e. its ramification index is equal to the degree of \(O\) over its coefficient ring.

IsUnramified(O): RngFunOrd -> BoolElt

Return whether a finite order \(O\) is unramified at the finite places and whether an infinite order \(O\) is unramified at the infinite places.

IsWildlyRamified(O): RngFunOrd -> BoolElt

Return whether there is a prime ideal of the order \(O\) which is wildly ramified, i.e. its ramification index is divisible by the characteristic of its residue class field.

IsInKummerRepresentation(K): FldFun -> BoolElt, FldFunElt

Tests if the global function field \(K\) is, in its current representation, a Kummer extension. More specific, this function tests if the defining polynomial is of the form \(x^r-a\) for some \(r\) coprime to the characteristic and if \(r\) divides the order of the multiplicative group of the constant field, ie. if the coefficient ring of \(K\) contains a primitive \(r\)-th root of unity. In case \(K\) is in Kummer representation, the element \(a\) is returned as a second return value.

IsInArtinSchreierRepresentation(K): FldFun -> BoolElt, FldFunElt

Tests if a global function field \(K\) is, in its current representation, a Artin-Schreier extension, ie. if the defining polynomial of \(K\) is of the form \(x^p-x-a\) where \(p\) is the characteristic of \(K\). In this case, the element \(a\) is returned as a second return value.