# Element Operations See also Section [Generic Element Functions](../IntroductionToRings/element.md#rngintro-elt-funcs). ## Arithmetic Operators ### `+ a: FldFinElt -> FldFinElt` ### `- a: FldFinElt -> FldFinElt` ### `a + b: FldFinElt, FldFinElt -> FldFinElt` ### `a - b: FldFinElt, FldFinElt -> FldFinElt` ### `a * b: FldFinElt, FldFinElt -> FldFinElt` ### `a / b: FldFinElt, FldFinElt -> FldFinElt` ### `a ^ k: FldFinElt, RngIntElt -> FldFinElt` ### `a +:= b: FldFinElt, FldFinElt -> FldFinElt` ### `a -:= b: FldFinElt, FldFinElt -> FldFinElt` ### `a *:= b: FldFinElt, FldFinElt -> FldFinElt` ## Equality and Membership ### `a eq b: FldFinElt, FldFinElt -> BoolElt` ### `a ne b: FldFinElt, FldFinElt -> BoolElt` ### `a in F: FldFinElt, Rng -> BoolElt` ### `a notin F: FldFinElt, Rng -> BoolElt` ## Parent and Category ### `Parent(a): FldFinElt -> FldFin` ### `Category(a): FldFinElt -> Cat` ## Predicates on Ring Elements ### `IsZero(a): FldFinElt -> BoolElt` ### `IsOne(a): FldFinElt -> BoolElt` ### `IsMinusOne(a): FldFinElt -> BoolElt` ### `IsNilpotent(a): FldFinElt -> BoolElt` ### `IsIdempotent(a): FldFinElt -> BoolElt` ### `IsUnit(a): FldFinElt -> BoolElt` ### `IsZeroDivisor(a): FldFinElt -> BoolElt` ### `IsRegular(a): FldFin -> BoolElt` ### `IsIrreducible(a): FldFinElt -> BoolElt` ### `IsPrime(a): FldFinElt -> BoolElt` ### `IsPrimitive(a): FldFinElt -> BoolElt` Returns `true` if and only if the element $a$ of $F$ is a primitive element for $F$ (i.e., if and only if the multiplicative order of $a$ is $\#F-1$). ### `IsPrimitive(f): RngUPolElt -> BoolElt` Given a univariate polynomial $f\in F[x]$, over a finite field $F$, such that the degree of $f$ is greater than or equal to 1, this function returns `true` if and only if $f$ defines a primitive extension $G=F[x]/f$ of $F$ (that is, $x$ is primitive in $G$). ### `IsNormal(a): FldFinElt -> BoolElt` Returns `true` if and only if the element $a$ of $F$ generates a normal basis for the field over the ground field, that is, if and only if $a, a^q, \ldots, a^{q^{n-1}}$ form a basis for $F$ over the ground field $G={\bf F}_{q}$. ### `IsNormal(a, E): FldFinElt -> BoolElt` Returns `true` if and only if the element $a$ of the finite field $F$ with $q^n$ elements generates a normal basis for $F$ over its subfield $E$, that is, if and only if $a, a^q, \ldots, a^{q^{n-1}}$ form a basis for $F$ over $E$ for $q=\#E$. ### `IsSquare(a): FldFinElt -> BoolElt` Given a finite field element $a\in F$, this function returns either `true` and an element $b\in F$ such that $b^2=a$, or it returns `false` in the case that such an element does not exist. ## Minimal and Characteristic Polynomial ### `MinimalPolynomial(a): FldFinElt -> RngUPolElt` The minimal polynomial of the element $a$ of the field $F$, relative to the ground field of $F$. This is the unique minimal-degree monic polynomial with coefficients in the ground field, having $a$ as a root. ### `MinimalPolynomial(a, E): FldFinElt, FldFin -> RngUPolElt` The minimal polynomial of the element $a$ of the field $F$, relative to the subfield $E$ of $F$. This is the unique minimal-degree monic polynomial with coefficients in $E$, having $a$ as a root. ### `CharacteristicPolynomial(a): FldFinElt -> RngUPolElt` Given an element $a$ of a finite field $F$, return the characteristic polynomial of $a$ with respect to the ground field of $F$. (This polynomial is the characteristic polynomial of the companion matrix of $a$ written as a polynomial over the ground field, and is a power of the minimal polynomial.) ### `CharacteristicPolynomial(a, E): FldFinElt, FldFin -> RngUPolElt` Given an element $a$ of a finite field $F$, return the characteristic polynomial of $a$ with respect to the subfield $E$ of $F$. (This polynomial is the characteristic polynomial of the companion matrix of $a$ written as a polynomial over $E$, and is a power of the minimal polynomial over $E$.) ## Norm, Trace and Frobenius ### `Norm(a): FldFinElt -> FldFinElt` The norm of the element $a$ from the field $F$ to the ground field of $F$. ### `Norm(a, E): FldFinElt, FldFin -> FldFinElt` The relative norm of the element $a$ from the field $F$, with respect to the subfield $E$ of $F$. The result is an element of $E$. ### `AbsoluteNorm(a): FldFinElt -> FldFinElt` ### `NormAbs(a): FldFinElt -> FldFinElt` The absolute norm of the element $a$, that is, the norm to the prime subfield of the parent field $F$ of $a$. ### `Trace(a): FldFinElt -> FldFinElt` The trace of the element $a$ from the field $F$ to the ground field of $F$. ### `Trace(a, E): FldFinElt, FldFin -> FldFinElt` The relative trace of the element $a$ from field $F$, with respect to the subfield $E$ of $F$. The result is an element of $E$. ### `AbsoluteTrace(a): FldFinElt -> FldFinElt` ### `TraceAbs(a): FldFinElt -> FldFinElt` The trace of the element $a$, that is, the trace to the prime subfield of the parent field $F$ of $a$. ### `Frobenius(a): FldFinElt -> FldFinElt` The Frobenius image of $a$ w.r.t. the ground field of $K$; i.e., $a^{\#G}$, where $G$ is the ground field of the parent of $a$. ### `Frobenius(a, r): FldFinElt, RngIntElt -> FldFinElt` The $r$-th Frobenius image of $a$ w.r.t. the ground field of $K$; i.e., $a^{(\#G)^r}$, where $G$ is the ground field of the parent of $a$. ### `Frobenius(a, E): FldFinElt, FldFin -> FldFinElt` The Frobenius image of x w.r.t. $E$; i.e., $x^{\#E}$. ### `Frobenius(a, E, r): FldFinElt, FldFin, RngIntElt -> FldFinElt` The Frobenius image of x w.r.t. $E$; i.e., $x^{(\#E)^r}$. ### `NormEquation(K, y): FldFin, FldFin -> BoolElt, FldFinElt` Given a finite field $K$ and an element $y$ of a subfield $S$ of $K$, return whether an element $x\in K$ exists such that Norm$(x, S) = y$, and, if so, such an element $x$ (in $K$). ### `Hilbert90(a, q): FldFinElt, RngIntElt -> FldFinElt` Given an element $a$ of some finite field $k$ and a power $q$ of the characteristic of $k$, return a solution of the Hilbert 90 equation $x^qx^{-1}=a$. Note that the solution may be in a finite-degree extension of $k$. ### `AdditiveHilbert90(a, q): FldFinElt, RngIntElt -> FldFinElt` Given an element $a$ of some finite field $k$ and a power $q$ of the characteristic of $k$, return a solution of the additive Hilbert 90 equation $x^q - x=a$. Note that the solution may be in a finite-degree extension of $k$. ## Order and Roots ### `Order(a): FldFinElt -> RngIntElt` The multiplicative order of the non-zero element $a$ of the field $F$. ### `FactoredOrder(a): FldFinElt -> RngIntElt` The multiplicative order of the non-zero element $a$ of the field $F$ as a factorization sequence. ### `SquareRoot(a): FldFinElt -> FldFinElt` ### `Sqrt(a): FldFinElt -> FldFinElt` The square root of the non-zero element $a$ from the field $F$, i.e., an element $y$ of $F$ such that $y^2 = a$. An error results if $a$ is not a square. ### `Root(a, n): FldFinElt, RngIntElt -> FldFinElt` The $n$-th root of the non-zero element $a$ from the field $F$, i.e., an element $y$ of $F$ such that $y^n = a$. An error results if no such root exists. ### `IsPower(a, n): FldFinElt, RngIntElt -> BoolElt, FldFinElt` Given a finite field element $a\in F$, and an integer $n>0$, this function returns either `true` and an element $b\in F$ such that $b^n=a$, or it returns `false` in the case that such an element does not exist. ### `AllRoots(a, n): FldFinElt, RngIntElt -> SeqEnum` Given a finite field element $a\in F$, and an integer $n>0$, return a sequence containing all of the $n$-th roots of $a$ which lie in the same field $F$. ### `Example: Functions (ex-ead4e7)` Given the fields $F$ and $F49$ defined above, we can use the following functions: ```magma > F7 := FiniteField(7); > F49 := ext< F7 | 2 >; > F := ext< F49 | 2 >; > Root(z^73, 7); z^1039 > Trace(z^73); 1 > Trace(z^73, F49); w^44 > Norm(z^73); 3 > Norm(z^73, F49); w^37 > Norm(w^37); 3 > MinimalPolynomial(z^73); x^2 + w^20*x + w^43 > MinimalPolynomial(z^73, F7); x^4 + 4*x^2 + 4*x + 3 ``` We now demonstrate the `NormEquation` function. ```magma > Norm(z); 3 > NormEquation(F, F7!3); true z > Norm(z^30, F49); w^30 > Parent(z) eq F; true > NormEquation(F, w^30); true z^30 ```