Subsections

• Creation of Cyclotomic Fields
• Creation of Elements

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).

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

Given a sequence of cyclotomic field elements s, this function returns the smallest cyclotomic field (possibly the rational field) G containing each of those.

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

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.

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, 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.

ElementToSequence(a) : FldCycElt -> [ FldRatElt ]

Given an element a=alpha_0 + alpha_1zeta_m + ... + alpha_(m - 1)zeta_m^(m - 1) of the cyclotomic field Q(zeta_m), return the sequence [alpha_0, ..., alpha_(m - 1)] of rational coefficients. Note that this is a sequence of length m, but the coefficients are with respect to the internal integral basis, and as a consequence certain coefficients (such as alpha_(m - 1)) will always be zero.

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.