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Creation Functions

Creation Functions

Subsections

Creation of Cyclotomic Fields

CyclotomicField(m) : RngIntElt -> FldCyc
Given a positive integer m > 2, create the field obtained by adjoining the m-th roots of unity to Q. It is possible to assign a name to the primitive m-th root of unity zeta_(m) using angle brackets: R<s> := CyclotomicField(m).
CyclotomicPolynomial(m) : RngIntElt -> RngUPolElt
Given a positive integer m > 2, create the cyclotomic polynomial of order m. This function is just equivalent to DefiningPolynomial(CyclotomicPolynomial(m)).
MinimalField(a) : FldCycElt -> FldCyc
Given an element a from a cyclotomic field F, this function returns the smallest cyclotomic field (possibly the rational field) E subset F containing a.
MinimalField(S) : [ FldCycElt ] -> FldCyc
MinimalField(S) : { FldCycElt } -> FldCyc
Given a set or sequence S of cyclotomic field elements, this function returns the smallest cyclotomic field (possibly the rational field) G containing each of the elements of S.

Creation of Elements

For elements of cyclotomic number fields the following conventions are used. Primitive roots of unity zeta_m are chosen in such a way that zeta_m^(m/d)=zeta_d, for every divisor d of m; one may think of this as choosing zeta_m=( e)^(2pi i/m) in the complex plane for every m (a convention that is followed for the explicit embedding in the complex domains).

One(L) : FldCyc -> FldCycElt
Identity(L) : FldCyc -> FldCycElt
Zero(L) : FldCyc -> FldCycElt
Representative(L) : FldCyc -> FldCycElt
K ! [a_0, ..., a_m - 1] : FldCyc, [FldCycElt] -> FldCycElt
elt< K | a_0, ..., a_m - 1> : FldCyc, FldRatElt , FldRatElt -> FldCycElt
Given the cyclotomic field K=Q(zeta_m) and rational numbers a_0, a_(1), ..., a_(m - 1), construct the element a_0 + a_1zeta_(m) + ... + a_(m - 1)zeta_m^(m - 1) in K.
RootOfUnity(n) : RngIntElt, FldCyc -> FldCycElt
Create the n-th root of unity zeta_n in Q(zeta_n).
RootOfUnity(n, K) : RngIntElt, FldCyc -> FldCycElt
Given a cyclotomic field Q(zeta_m) and an integer n>2, create the n-th root of unity zeta_n in K. An error results if zeta_n notin K, that is, if n does not divide m (or 2m in case m is odd).
K ! a : FldCyc, RngIntElt -> FldCycElt
K ! a : FldCyc, FldRatElt -> FldCycElt
Coerce a into the cyclotomic field K; this will work for any integer or rational number, as well as for those elements from quadratic or cyclotomic fields that are in K.
Eltseq(a) : FldCycElt -> [ FldRatElt ]
ElementToSequence(a) : FldCycElt -> [ FldRatElt ]
Given an element a=alpha_0 + alpha_1zeta_m + ... + alpha_(d - 1)zeta_m^(d - 1) of the cyclotomic field Q(zeta_m), where d=phi(m) is the degree of the field, return the sequence [alpha_0, ..., alpha_(d - 1)] of rational coefficients.
Minimise(~a) : FldCycElt ->
Minimize(~a) : FldCycElt ->
Given an element a in a cyclotomic field F, this procedure finds the minimal cyclotomic subfield E subset F containing a, and coerces a into E. Note that E may be Q.
Minimise(~s) : [ FldCycElt ] ->
Minimize(~s) : [ FldCycElt ] ->
Given a set s of cyclotomic field elements, this procedure finds the minimal cyclotomic field E containing all of them, and coerces each element into E. The resulting set will have universe E. Note that E may be Q.
Minimise(a) : FldCycElt -> RngElt
Minimize(a) : FldCycElt -> RngElt
Given an element a in a cyclotomic field F, this function finds the minimal cyclotomic subfield E subset F containing a, and coerces a into E. Note that E may be Q.
Minimise(s) : { FldCycElt } -> { RngElt }
Minimize(s) : { FldCycElt } -> { RngElt }
Given a set s of cyclotomic field elements, this function finds the minimal cyclotomic field E containing all of them, and coerces each element into E. The resulting set will have universe E. Note that E may be Q.
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