# Elements ## `Parent(x): AlgEtQElt -> AlgEtQ` ## `Algebra(x): AlgEtQElt -> AlgEtQ` Returns the algebra to which the element $x$ belongs to. ## `Components(x): AlgEtQElt -> SeqEnum` Given an element $x$, returns its components, which are elements of number fields. ## `AbsoluteCoordinates(x): AlgEtQElt -> SeqEnum` Given an element $x$, returns the coordinates relative to the absolute basis, which are elements of the prime rational field. ## `AbsoluteCoordinates(x, S): AlgEtQElt, AlgEtQOrd -> SeqEnum` Given an element $x$ and an order $S$, returns the coordinates of $x$ with respect to the stored Z-basis of $S$. ## `IsCoercible(A, x): AlgEtQ, Any -> BoolElt, AlgEtQElt` Return whether the element $x$ is coercible into the algebra $A$ and the result of the coercion if so. ## `A ! x: AlgEtQ, Any) -> AlgEtQElt` Coerce $x$ into the algebra $A$. ## `One(A): AlgEtQ -> AlgEtQElt` The multiplicative neutral element of the algebra $A$. ## `Zero(A): AlgEtQ -> AlgEtQElt` The additive neutral element of the algebra $A$. ## `IsUnit(x): AlgEtQElt -> BoolElt` Returns whether the element $x$ is a unit in its algebra $A$. ## `IsZeroDivisor(x): AlgEtQElt -> BoolElt` Returns whether the element $x$ is a zero-divisor in its algebra $A$. ## `Random(A, bd): AlgEtQ, RngIntElt -> AlgEtQElt` Returns a random element of the algebra $A$. The coefficients are bounded by the positive integer $bd$. ## `Random(A): AlgEtQ -> AlgEtQElt` ```magma bd: RngIntElt Default: 3 ``` Returns a random element of the algebra $A$. The coefficients are bounded by the parameter $bd$ (default 3). ## `RandomUnit(A, bd): AlgEtQ, RngIntElt -> AlgEtQElt` Returns a random unit of the algebra $A$. The coefficients are bounded by the positive integer $bd$. ## `x1 eq x2: AlgEtQElt, AlgEtQElt -> BoolElt` ## `x1 eq x2: RngIntElt, AlgEtQElt -> BoolElt` ## `x1 eq x2: AlgEtQElt, RngIntElt -> BoolElt` ## `x1 eq x2: FldRatElt, AlgEtQElt -> BoolElt` ## `x1 eq x2: AlgEtQElt, FldRatElt -> BoolElt` Returns whether the elements $x1$ and $x2$ are equal. ## `x1 + x2: AlgEtQElt, AlgEtQElt -> AlgEtQElt` ## `x1 + x2: Any, AlgEtQElt -> AlgEtQElt` ## `x1 + x2: AlgEtQElt, Any -> AlgEtQElt` ## `- x: AlgEtQElt -> AlgEtQElt` ## `x1 - x2: AlgEtQElt, AlgEtQElt -> AlgEtQElt` ## `x1 - x2: Any, AlgEtQElt -> AlgEtQElt` ## `x1 - x2: AlgEtQElt, Any -> AlgEtQElt` ## `x1 * x2: AlgEtQElt, AlgEtQElt -> AlgEtQElt` ## `x1 * x2: Any, AlgEtQElt -> AlgEtQElt` ## `x1 * x2: AlgEtQElt, Any -> AlgEtQElt` ## `Inverse(x): AlgEtQElt -> AlgEtQElt` The multiplicative inverse of the algebra element $x$. ## `x ^ n: AlgEtQElt, RngIntElt -> AlgEtQElt` ## `x1 / x2: AlgEtQElt, AlgEtQElt -> AlgEtQElt` ## `x1 / x2: Any, AlgEtQElt -> AlgEtQElt` ## `x1 / x2: AlgEtQElt, Any -> AlgEtQElt` ## `&+ seq: SeqEnum[AlgEtQElt] -> AlgEtQElt` Given a sequence of `AlgEtQElt` returns the sum of the entries. ## `&* seq: SeqEnum[AlgEtQElt] -> AlgEtQElt` Given a sequence of `AlgEtQElt` returns the product of the entries. ## `DotProduct(a, b): SeqEnum, SeqEnum -> Any` Given two sequences $a=[a_1,\ldots,a_n]$ and $b=[b_1,\ldots,b_n]$, returns $\sum_i a_i\cdot b_i$. ## `Example: Dot Product Example (ex-1d7031)` ```magma > _:=PolynomialRing(Integers()); > f := (x^8+16)*(x^8+81); > A := EtaleAlgebra(f); > // We compute the `canonical` primitive element, which is the class of the > // variable x in A. > a := PrimitiveElement(A); a; <$.1, $.1> > // The algebra A has two components: > comps, embeddings, projections:=Components(A); > K1, K2 := Explode(comps); > // The unit element of each component corresponds to an orthogonal idempotent > // of A: > [ embeddings[1](K1!1),embeddings[2](K2!1) ] eq OrthogonalIdempotents(A); true > // We conclude this example by showing the use of DotProduct and > // its timings advantages: > N := 10^5; > elts1 := [ a+i : i in [1..N] ]; > elts2 := [ a-i : i in [1..N] ]; > time s1 := &+[ elts1[i]*elts2[i] : i in [1..N] ]; Time: 0.450 > time s2 := DotProduct(elts1,elts2); Time: 0.150 > s1 eq s2; true ``` ## `MinimalPolynomial(x): AlgEtQElt -> RngUPolElt` Returns the minimal polynomial over the common base ring of the number fields defining the algebra $A$ of the element $x$. ## `MinimalPolynomial(x, F): AlgEtQElt, Rng -> RngUPolElt` Returns the minimal polynomial over the ring $F$ of the element $x$. ## `AbsoluteMinimalPolynomial(x): AlgEtQElt -> RngUPolElt` Returns the minimal polynomial over the prime field of the element $x$ or an algebra. ## `IsIntegral(x): AlgEtQElt -> BoolElt` Returns whether the element $x$ of an algebra is integral (over the integers). ## `Evaluate(f, a): RngUPolElt, AlgEtQElt -> AlgEtQElt` Evaluate the polynomial $f$ at the algebra element $a$. ## `PrimitiveElement(A): AlgEtQ -> AlgEtQElt` Returns the primitive element of the étale algebra $A$. Note that $A$ has a primitive element only if it is the product of distinct number fields. Given an étale algebra $A$ over ${{\Bbb Q}}$ there exists an element $a\in A$ such that $A = {{\Bbb Q}}[a]$, that is, every element can be written as a polynomial with rational coefficients in $a$. Such an element is called a *primitive element* of $A$. It is characterized by having a minimal polynomial whose degree equals the absolute dimension of $A$. The intrinsic `PrimitiveElement` produces such an element of the étale algebra $A$ using a deterministic procedure which we now describe: Let $N$ be the number of components of $A$, each one having primitive element $a_i$. Set $b_1$ = $a_1$. For $i=2,\ldots,N$, set $b_i = a_i+j$ where $j$ is the smallest non-negative integer such that the minimal polynomial of $a_i+j$ is not in the set of minimal polynomials of the elements $b_1,\ldots,b_{i-1}$. The output is the element of $A$ whose components are $b_1,...,b_N$. In particular, if $A$ is a product of number fields with different defining polynomials, then the output is the element of $A$ whose components are the primitive elements of the components. ## `PowerBasis(A): AlgEtQ -> SeqEnum[AlgEtQElt]` Returns the power basis of the étale algebra $A$, consisting of powers of the primitive element of $A$. ## `Basis(A): AlgEtQ -> SeqEnum` Returns a basis of the algebra $A$ over the common base field. ## `AbsoluteBasis(A): AlgEtQ -> SeqEnum` Returns a basis of the algebra $A$ over the prime field. ## `A . i: AlgEtQ, RngIntElt -> AlgEtQElt` Returns the $i$-th element of the absolute basis of $A$. ## `AbsoluteCoordinates(seq, basis): SeqEnum[AlgEtQElt], SeqEnum[AlgEtQElt] -> SeqEnum` Given a sequence of elements and a basis over the `PrimeField` returns a sequence whose entries are the coordinates in the `PrimeField` with respect to the given basis. ## `OrthogonalIdempotents(A): AlgEtQ -> SeqEnum` Returns the orthogonal idempotent element of the étale algebra $A$. ## `Idempotents(A): AlgEtQ -> SeqEnum` Returns the idempotent element of the étale algebra $A$.