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
Algebra(x) : AlgEtQElt -> AlgEtQ
Returns the algebra to which the element x belongs to.
Given an element x, returns its components, which are elements of number fields.
Given an element x, returns the coordinates relative to the absolute basis, which are elements of the prime rational field.
Return whether the element x is coercible into the algebra A and the result of the coercion if so.
Coerce x into the algebra A.
The multiplicative neutral element of the algebra A.
The additive neutral element of the algebra A.
Returns whether the element x is a unit in its algebra A.
Returns whether the algebra element x is a not unit in its algebra A.
Returns a random element of the algebra A. The coefficients are bounded by the positive integer bd.
bd: RngIntElt Default: 3
Returns a random element of the algebra A. The coefficients are bounded by the parameter bd (default 3).
Returns a random unit of the algebra A. The coefficients are bounded by the positive integer bd.
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.
The multiplicative inverse of the algebra element x.
Given a sequence of AlgEtQElt returns the sum of the entries.
Given a sequence of AlgEtQElt returns the product of the entries.
SumOfProducts(as, bs) : SeqEnum[RngIntElt], SeqEnum[AlgEtQElt] -> AlgEtQElt
SumOfProducts(as, bs) : SeqEnum[AlgEtQElt], SeqEnum[RngIntElt] -> AlgEtQElt
SumOfProducts(as, bs) : SeqEnum[FldRatElt], SeqEnum[AlgEtQElt] -> AlgEtQElt
SumOfProducts(as, bs) : SeqEnum[AlgEtQElt], SeqEnum[FldRatElt] -> AlgEtQElt
Given sequences as and bs containing elements of algebras, of equal length, returns &+[as[i]*bs[i] : i in [1..#as]]. This intrinsic is included to obviate to the loss of efficiency due to the many calls of IsCoercible.
Returns the minimal polynomial over the common base ring of the number fields defining the algebra A of the element x.
Returns the minimal polynomial over the ring F of the element x.
Returns the minimal polynommial over the prime field of the element x or an algebra.
Returns whether the element x of an algebra is integral (over the integers).
Evaluate the polynomial f at the algebra element a.
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.
Returns the power basis of the étale algebra A, consisting of powers of the primitive element of A.
Returns a basis of the algebra A over the common base field.
Returns a basis of the algebra A over the prime field.
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.
Returns the orthogonal idempotent element of the étale algebra A.
Returns the idempotent element of the étale algebra A.
V2.29, 21 October 2025