# 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` ```magma 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` ```magma 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 := 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 := 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. ```magma > C := 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 ``` ## 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: 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: 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. ```magma > 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 ``` 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`