Creation Functions#

We describe the creation of real and complex fields and their elements.

Creation of Structures#

At the time Magma is loaded, a real field is automatically created. This is used as the default parent for literal reals and real values returned by Magma.

RealField(p): RngIntElt -> FldRe#

Bits: BoolElt                    Default: false

Given a positive integer \(p\), create and return a version \(R\) of the real field \({\mathbb{R}}\) in which all calculations are correct to precision \(p\). If the parameter Bits is true, then the precision \(p\) is specified as the number of binary digits. If Bits is false, then the precision is given as the number of decimal digits — this is translated into a binary precision of \(\lceil \log_2 {10^p} \rceil\).

RealField() -> FldRe#: Return the default real field.

ComplexField(p): RngIntElt -> FldCom#

Bits: BoolElt                    Default: false

Given a positive integer \(p\), create and return a version \(C\) of the complex field \({\mathbb{C}}\) in which all calculations are correct to precision \(p\). If the parameter Bits is true, then the precision \(p\) is specified as the number of binary digits. If Bits is false, then the precision is given as the number of decimal digits — this is translated into a binary precision of \(\lceil \log_2 {10^p} \rceil\). By default no name is given to \(\sqrt{-1}\); this may be changed with AssignNames. Angle brackets, e.g. C<i> := ComplexField(20), may be used to assign \(\sqrt{-1}\) to an identifier.

ComplexField() -> FldCom#: Return the default complex field. By default no name is given to \(\sqrt{-1}\); this may be changed with AssignNames. Angle brackets, e.g. C<i> := ComplexField(), may be used to assign \(\sqrt{-1}\) to an identifier.

ComplexField(R): FldRe -> FldCom#: Return the complex field which has real subfield \(R\); in other words, return the complex field with the same precision as the real field \(R\).

Example: Create Complex Field (ex-108c46)#

It is convenient to use \({\rm i}\) to define elements of a complex field. It is also possible to change the default printing of i, using AssignNames, as follows. Note that the latter procedure does not assign to an identifier, it only changes the printing.

> C<i> := ComplexField(20);
> Pi(C)+ 1/4*i;
3.1415926535897932385 + 0.25000000000000000000*i
> AssignNames(~C, ["k"]);
> Pi(C)+ 1/4*i;
3.1415926535897932385 + 0.25000000000000000000*k
> k := Name(C, 1);
> Pi(C)+ 1/4*k;
3.1415926535897932385 + 0.25000000000000000000*k

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Creation of Elements#

a . becpd: RngIntElt, RngIntElt, RngIntElt -> FldReElt#
a . bEcPd: RngIntElt, RngIntElt, RngIntElt -> FldReElt#
a.bec P d: RngIntElt, RngIntElt, RngIntElt -> FldReElt#
a.bec p d: RngIntElt, RngIntElt, RngIntElt -> FldReElt#
a.b E cpd: RngIntElt, RngIntElt, RngIntElt -> FldReElt#
a.b e cpd: RngIntElt, RngIntElt, RngIntElt -> FldReElt#: Given a succession of literal decimal digits \(a\), a succession of literal decimal digits \(b\), a succession of literal decimal digits \(c\), and an integer \(d\), construct the real number \(r=a.b \times 10^c\). If specified, the effect of \(d\) is to create \(r\) as an element of the real field of precision \(d\). If \(d\) is omitted (together with p or P), the real number will be created as an element of the default real field. Both \(a\) and \(c\) may include a leading sign + or -; leading zeroes in \(a\) and \(c\) are ignored. If \(b\) consists entirely of zeroes it may be omitted together with the . and if \(c\) is zero it may be omitted together with E (or e). But note that if all of \(b\), \(c\) and \(d\) are omitted the result will be an integer.

elt<R | m, n>: FldRe, FldReElt, RngIntElt -> FldReElt#: Given the real field \(R\), an element \(m\) coercible into \(R\) and an integer \(n\), construct the real number \(m \times 2^n\) in \(R\).

elt<C | x, y>: FldCom, FldReElt, FldReElt -> FldComElt#
C ! [x, y]: FldCom, [FldReElt, FldReElt] -> FldComElt#: Given the complex field \(C\) and elements \(x\) and \(y\) coercible into the real field underlying \(C\), construct the complex number \(x+y{\mathrm{i}}\).

R ! a: FldRe, RngElt -> FldReElt#

Given an integer, a rational number, a quadratic or cyclotomic number field element \(a\), this returns an element from the real field \(R\) that best approximates \(a\). An error results if \(a\) is a non-real quadratic or cyclotomic field element. If \(R\) is a field of precision \(r\) and \(a\) is an element of a real field \(S\) of precision \(s\) then:

•: if \(a\) is an element of a real field \(S\) of precision \(s \geq r\), then an element of \(R\) approximating \(a\) to \(r\) digits is returned;
•: if \(a\) is an element of a real field \(S\) of precision \(s < r\), then an element of \(R\) is returned approximating \(a\), obtained by padding with zeroes until the required precision \(r\) is reached;

C ! a: FldCom, RngElt -> FldComElt#: Given an integer, a rational number, a quadratic or cyclotomic number field element \(a\), this returns an element from the complex field \(C\) that best approximates \(a\). The rules of coercion for the real and imaginary parts are the same as those for coercion into a real field.

Example: Create Elements (ex-356721)#

We create the real number \(1.2345\) in many ways. We assume that the default real field has not been changed.

> x := 1.2345;
> x, Parent(x);
1.23450000000000000000000000000 Real field of precision 30
> SetDefaultRealField(RealField(20));
> x1 := 1.2345;
> x1, Parent(x1);
1.2345000000000000000 Real field of precision 20
> x2 := 12345e-4;
> x2, Parent(x2);
1.2345000000000000000 Real field of precision 20
> x3 := 1.2345p10;
> x3, Parent(x3);
1.234500000 Real field of precision 10
> x4 := 12345e-4p8;
> x4, Parent(x4);
1.2345000 Real field of precision 8
> x5 := RealField(12) ! 1.2345;
> x5, Parent(x5);
1.23450000000 Real field of precision 12

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The following generic element constructions are available; they return the \(1\) and \(0\) element of a real or complex field, where any zero elements are the “positive zero” in MPFR.

One(R): FldRe -> FldReElt#

One(R): FldCom -> FldComElt#

Identity(R): FldRe -> FldReElt#

Identity(R): FldCom -> FldComElt#

Zero(R): FldRe -> FldReElt#

Zero(R): FldCom -> FldComElt#

Representative(R): FldRe -> FldReElt#

Representative(R): FldCom -> FldComElt#