# Element Operations A variety of different types of operations are provided for rational elements including arithmetic operations, comparison and predicates and converting to a sequence. ## Parent and Category ### `Parent(r): FldRatElt -> FldRat` ### `Category(r): FldRatElt -> Cat` ## Arithmetic Operators ### `+ a: FldRatElt -> FldRatElt` ### `- a: FldRatElt -> FldRatElt` ### `a + b: FldRatElt, FldRatElt -> FldRatElt` ### `a - b: FldRatElt, FldRatElt -> FldRatElt` ### `a * b: FldRatElt, FldRatElt -> FldRatElt` ### `a ^ k: FldRatElt, RngIntElt -> FldRatElt` ### `a / b: FldRatElt, FldRatElt -> FldRatElt` ### `a +:= b: FldRatElt, FldRatElt -> FldRatElt` ### `a -:= b: FldRatElt, FldRatElt -> FldRatElt` ### `a *:= b: FldRatElt, FldRatElt -> FldRatElt` ### `a /:= b: FldRatElt, FldRatElt -> FldRatElt` ### `a ^:= k: FldRatElt, RngIntElt -> FldRatElt` ## Numerator and Denominator ### `Numerator(q): FldRatElt -> RngIntElt` The (integer) numerator of the rational number $q$ in reduced form. ### `Denominator(q): FldRatElt -> RngIntElt` The (integer) denominator of the rational number $q$ in reduced form. This will always be a positive integer. ### `Example: numerator (ex-da083f)` Rational numbers are always immediately put in reduced form, that is, the greatest common divisor of numerator and denominator is taken out, and the denominator will be positive. ```magma > Numerator(10/-4); -5 > Denominator(10/-4); 2 ``` ## Equality and Membership ### `a eq b: FldRatElt, FldRatElt -> BoolElt` ### `a ne b: FldRatElt, FldRatElt -> BoolElt` ### `a in R: FldRatElt, Rng -> BoolElt` ### `a notin R: FldRatElt, Rng -> BoolElt` ## Predicates on Ring Elements ### `IsIntegral(q): FldRatElt -> BoolElt` Returns `true` if the rational number $q$ is an element of the ring of integers, `false` otherwise. ### `IsZero(a): FldRatElt -> BoolElt` ### `IsOne(a): FldRatElt -> BoolElt` ### `IsMinusOne(a): FldRatElt -> BoolElt` ### `IsNilpotent(a): FldRatElt -> BoolElt` ### `IsIdempotent(a): FldRatElt -> BoolElt` ### `IsUnit(a): FldRatElt -> BoolElt` ### `IsZeroDivisor(a): FldRatElt -> BoolElt` ### `IsRegular(a): FldRatElt -> BoolElt` ### `IsIrreducible(a): FldRatElt -> BoolElt` ### `IsPrime(a): FldRatElt -> BoolElt` ## Comparison ### `a gt b: FldRatElt, FldRatElt -> BoolElt` ### `a ge b: FldRatElt, FldRatElt -> BoolElt` ### `a lt b: FldRatElt, FldRatElt -> BoolElt` ### `a le b: FldRatElt, FldRatElt -> BoolElt` ### `Maximum(a, b): FldRatElt, FldRatElt -> FldRatElt` ### `Maximum(Q): [FldRatElt] -> FldRatElt` ### `Minimum(a, b): FldRatElt, FldRatElt -> FldRatElt` ### `Minimum(Q): [FldRatElt] -> FldRatElt` ## Conjugates, Norm and Trace ### `ComplexConjugate(q): FldRatElt -> FldRatElt` The complex conjugate of $q$, which will be the rational number $q$ itself. ### `Conjugate(q): FldRatElt -> FldRatElt` The conjugate of $q$, which will be the rational number $q$ itself. ### `Norm(q): FldRatElt -> FldRatElt` The norm (in ${\mathbb{Q}}$) of $q$, which will be the rational number $q$ itself. ### `Trace(q): FldRatElt -> FldRatElt` The trace (in ${\mathbb{Q}}$) of $q$, which will be the rational number $q$ itself. ### `MinimalPolynomial(q): FldRatElt -> RngUPolElt` Returns the minimal polynomial of the rational number $q$, which is the monic linear polynomial with constant coefficient $q$ in a univariate polynomial ring $R$ over the rational field. (If $R$ has not been created before with a name for its indeterminate, `$.1-q` will be returned.) ## Absolute Value and Sign ### `AbsoluteValue(q): FldRatElt -> FldRatElt` ### `Abs(q): FldRatElt -> FldRatElt` The absolute value $\vert q\vert$ of a rational number $q$. ### `Sign(q): FldRatElt -> RngIntElt` Returns the sign of the rational number $q$, which is one of the integers $-1$, $0$, $1$, corresponding to the cases $q<0$, $q=0$, and $q>0$. ### `Height(q): FldRatElt -> RngIntElt` The height of $q=r/s$. For $r$ and $s$ coprime, the height is defined as the maximum of the absolute value of $r$ and $s$. ## Rounding and Truncating ### `Ceiling(q): FldRatElt -> RngIntElt` The ceiling of the rational number $q$, that is, the least integer greater than or equal to $q$. ### `Floor(q): FldRatElt -> RngIntElt` The floor of the rational number $q$, that is, the largest integer less than or equal to $q$. ### `Round(q): FldRatElt -> RngIntElt` This function returns the integer value of the rational number $q$ rounded to the nearest integer. In the case of a tie, rounding is done away from zero (that is, $i+{1\over2}$ is rounded to $i+1$, for non-negative integers $i$ and $i-{1\over2}$ is rounded to $i-1$, for non-positive integers $i$). ### `Truncate(q): FldRatElt -> RngIntElt` This function returns the integer truncation of the rational number $q$, that is the integral part of $q$. Thus the effect is that of rounding towards $0$. ### `Qround(q, M): FldRatElt, RngIntElt -> FldRatElt` ```magma ContFrac: BoolElt Default: false ``` Finds a rational approximation $d$ of $q$ such that the denominator of $d$ is bounded by $M$. If `ContFrac` is given then an optimal approximation is computed using the continued fraction process. By default $d$ is obtained by some rounding procedure which is faster but gives worse results. ## Continued Fractions ### `ContinuedFraction(r): FldRatElt -> [ RngIntElt ]` Given a rational $r$, return the sequence of partial quotients of the continued fraction expansion of $r$. ### `ContinuedFractionValue(C): [ RngIntElt ] -> FldRatElt` Given a continued fraction expansion $C$, return the rational $r$ such that $C$ equals `ContinuedFraction(r)`. ### `HirzebruchJungContinuedFraction(r): FldRatElt -> [ RngIntElt ]` ### `HJContinuedFraction(r): FldRatElt -> [ RngIntElt ]` Given a rational $r$, return the sequence of partial quotients of the Hirzebruch-Jung continued fraction expansion of $r$. ### `HirzebruchJungContinuedFractionValue(C): [ RngIntElt ] -> FldRatElt` ### `HJContinuedFractionValue(C): [ RngIntElt ] -> FldRatElt` Given a Hirzebruch-Jung continued fraction expansion $C$, return the rational $r$ so that `HJContinuedFraction(r)` equals $C$. ## Rational Reconstruction Under certain circumstances it is useful to have a partial inverse of the function $\psi_m\colon{\mathbb{Q}}\rightarrow {\mathbb{Z}}/m{\mathbb{Z}}$ of taking residues modulo $m$ (where the obvious value of $\psi_m$ is only defined for rational numbers with denominator in smallest terms coprime to $m$); the partial inverse of the function is sometimes referred to as ‘rational reconstruction’. For $s\in {\mathbb{Z}}/m{\mathbb{Z}}$ the value of $\psi^{-1}(s)$ is the rational number $r$ for which $\psi_m(r)=s$ and, in addition, the absolute values of both the numerator and denominator of $r$ are at most $\sqrt{m/2}$; such $r$ does not always exist, but if $r$ exists it is unique. ### `RationalReconstruction(s): RngIntResElt -> BoolElt, FldRatElt` ### `RationalReconstruction(s): FldFinElt -> BoolElt, FldRatElt` Given an element $s$ of a ring $S$ of $m$ elements, return a Boolean flag indicating whether or not a rational number $r$ exists such that for the representation $r=n/d$ in minimal terms it holds that $n\cdot d^{-1}\equiv s\bmod m$, $\vert n\vert\le\sqrt{m/2}$ and $0 < d \le\sqrt{m/2}$. If the flag is true, the element $r$ is also returned. The ring $S$ is allowed to be a residue class ring `Integers(m)` or a finite field of prime cardinality $p=m$: `FiniteField(p)`. In addition, $s$ is allowed to be a matrix over a prime finite field, in which case the existence (and, if possible, value) of a rational reconstruction of the matrix is determined. ## Valuation ### `Valuation(x, p): FldRatElt, RngIntElt -> RngIntElt, FldRatElt` ### `Valuation(x, I): FldRatElt, RngIntElt -> RngIntElt, FldRatElt` The valuation $v$ of the rational number $x$ at the prime $p$ (the prime ideal $I$). This is the difference of the valuations of the numerator and denominator of $x$. The optional second return value is the rational $u$ such that $x = p^v u$. ## Sequence Conversions ### `ElementToSequence(a): FldRatElt -> [FldRatElt]` ### `Eltseq(a): FldRatElt -> [FldRatElt]` The sequence $[a]$ for compatibility with the other field types.