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Structure Operations

Structure Operations

In cyclotomic fields the generic ring functions listed below are supported. In the list below a and b are cyclotomic field elements; however, automatic coercion will ensure that +, *, -, /, eq, ne and in will also work if a or b is an integer or rational number.

Subsections

Related Structures

Parent(L) : FldCyc -> Pow
Category(L) : FldCyc -> FldCyc
IntegralBasis(K) : FldCyc -> [ FldCycElt ]
A sequence of elements of the cyclotomic field K that forms an integral basis for K. The integral bases for cyclotomic fields are chosen in such a way that they are compatible with each other, that is, if K is contained in L, then its integral basis will be contained in that for L.

Invariants

Characteristic(K) : FldCyc -> RngIntElt
Conductor(K) : FldCyc -> RngIntElt
The smallest n such that the field K is contained in Q(zeta_n); for a cyclotomic field that is either the `cyclotomic order' m (see below) or half that, depending on whether m not = 2 mod 4 or m = 2 mod 4.
Degree(K) : FldCyc -> RngIntElt
The (absolute) degree of K over Q. If K=Q(zeta_m) this equals phi(m).
Discriminant(K) : FldCyc -> RngIntElt
The discriminant of the cyclotomic field K.
CyclotomicOrder(K) : FldCyc -> RngIntElt
The value of m for the cyclotomic field Q(zeta_m). Note that this will be the m with which the cyclotomic field was created.
Signature(K) : FldCyc -> RngIntElt
Given a cyclotomic field K, return the signature of K, i.e. two integers, being the number of real embeddings and the number of pairs of complex embeddings of K.

Ring Predicates and Booleans

IsCommutative(Q) : FldCyc -> BoolElt
IsUnitary(Q) : FldCyc -> BoolElt
IsFinite(Q) : FldCyc -> BoolElt
IsOrdered(Q) : FldCyc -> BoolElt
IsField(Q) : FldCyc -> BoolElt
IsEuclideanDomain(Q) : FldCyc -> BoolElt
IsPID(Q) : FldCyc -> BoolElt
IsUFD(Q) : FldCyc -> BoolElt
IsDivisionRing(Q) : FldCyc -> BoolElt
IsEuclideanRing(Q) : FldCyc -> BoolElt
IsPrincipalIdealRing(Q) : FldCyc -> BoolElt
IsDomain(Q) : FldCyc -> BoolElt
K eq L : FldCyc , FldCyc -> BoolElt
K ne L : FldCyc , FldCyc -> BoolElt
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