Documentation

Elements

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

x ^ n : AlgEtQElt, RngIntElt -> AlgEtQElt

x1 / x2 : AlgEtQElt, AlgEtQElt -> AlgEtQElt

x1 / x2 : Any, AlgEtQElt -> AlgEtQElt

x1 / x2 : AlgEtQElt, Any -> AlgEtQElt

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

    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.

Inverse(x) : AlgEtQElt -> AlgEtQElt

The multiplicative inverse of the algebra element x.

&+ 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=[a₁, ..., a_n] and b=[b₁, ..., b_n], returns ∑_i a_i.b_i.

Example `AlgEtQ_DotProductExample (H42E4)`

> _<x>:=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 Q there exists an element a∈A such that A = 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 b₁ = a₁. For i=2, ..., 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 b₁, ..., b_{i - 1}. The output is the element of A whose components are b₁, ..., 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.

V2.29, 28 November 2025