Elements

  (O::NumFieldOrder)(a::NumFieldElem, check::Bool = true) -> NumFieldOrderElem

Given an element $a$ of the ambient number field of $\mathcal O$, this function coerces the element into $\mathcal O$. It will be checked that $a$ is contained in $\mathcal O$ if and only if check is true.

  (O::NumFieldOrder)(a::NumFieldOrderElem, check::Bool = true) -> NumFieldOrderElem

Given an element $a$ of some order in the ambient number field of $\mathcal O$, this function coerces the element into $\mathcal O$. It will be checked that $a$ is contained in $\mathcal O$ if and only if check is true.

  (O::NumFieldOrder)(a::IntegerUnion) -> NumFieldOrderElem

Given an element $a$ of type ZZRingElem or Integer, this function coerces the element into $\mathcal O$.

  (O::AbsNumFieldOrder)(arr::Vector{ZZRingElem})

Returns the element of $\mathcal O$ with coefficient vector arr.

  (O::AbsNumFieldOrder)(arr::Vector{Integer})

Returns the element of $\mathcal O$ with coefficient vector arr.

parent(a::NumFieldOrderElem) -> NumFieldOrder

Returns the order of which $a$ is an element.

parent(a::MatRingElem{T}) where T <: NCRingElement

Return the parent object of the given matrix.

elem_in_nf(a::NumFieldOrderElem) -> NumFieldElem

Returns the element $a$ considered as an element of the ambient number field.

coordinates(a::AbsNumFieldOrderElem) -> Vector{ZZRingElem}

Returns the coefficient vector of $a$ with respect to the basis of the order.

discriminant(B::Vector{NumFieldOrderElem})

Returns the discriminant of the family $B$ of algebraic numbers, i.e. $det((tr(B[i]*B[j]))_{i, j})^2$.

discriminant(E::EllipticCurve) -> FieldElem

Return the discriminant of $E$.

discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

discriminant(g::Vector)

Compute the product of all differences of distinct elements in the array.

==(x::NumFieldOrderElem, y::NumFieldOrderElem) -> Bool

Returns whether $x$ and $y$ are equal.

representation_matrix(a::AbsNumFieldOrderElem) -> ZZMatrix

Returns the representation matrix of the element $a$.

representation_matrix(a::AbsNumFieldOrderElem, K::AbsSimpleNumField) -> FakeFmpqMat

Returns the representation matrix of the element $a$ considered as an element of the ambient number field $K$. It is assumed that $K$ is the ambient number field of the order of $a$.

tr(a::NumFieldOrderElem)

Returns the trace of $a$ as an element of the base ring.

norm(a::NumFieldOrderElem)

Returns the norm of $a$ as an element in the base ring.

absolute_norm(a::NumFieldOrderElem) -> ZZRingElem

Return the absolute norm as an integer.

absolute_tr(a::NumFieldOrderElem) -> ZZRingElem

Return the absolute trace as an integer.

rand(O::AbsSimpleNumFieldOrder, n::IntegerUnion) -> AbsNumFieldOrderElem

Computes a coefficient vector with entries uniformly distributed in $\{-n,\dotsc,-1,0,1,\dotsc,n\}$ and returns the corresponding element of the order $\mathcal O$.

minkowski_map(a::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}

Returns the image of $a$ under the Minkowski embedding. Every entry of the array returned is of type ArbFieldElem with radius less then 2^-abs_tol.

conjugates_arb(x::NumFieldOrderElem, abs_tol::Int) -> Vector{AcbFieldElem}

Compute the conjugates of $x$ as elements of type AcbFieldElem. Recall that we order the complex conjugates $\sigma_{r+1}(x),...,\sigma_{r+2s}(x)$ such that $\sigma_{i}(x) = \overline{\sigma_{i + s}(x)}$ for $r + 2 \leq i \leq r + s$.

Every entry $y$ of the array returned satisfies radius(real(y)) < 2^-abs_tol, radius(imag(y)) < 2^-abs_tol respectively.

conjugates_arb_log(x::NumFieldOrderElem, abs_tol::Int) -> Vector{ArbFieldElem}

Returns the elements $(\log(\lvert \sigma_1(x) \rvert),\dotsc,\log(\lvert\sigma_r(x) \rvert), \dotsc,2\log(\lvert \sigma_{r+1}(x) \rvert),\dotsc, 2\log(\lvert \sigma_{r+s}(x)\rvert))$ as elements of type ArbFieldElem radius less then 2^-abs_tol.

t2(x::NumFieldOrderElem, abs_tol::Int = 32) -> ArbFieldElem

Return the $T_2$-norm of $x$. The radius of the result will be less than 2^-abs_tol.

minpoly(M::MatElem{T}, charpoly_only::Bool = false) where {T <: RingElement}
minpoly(S::PolyRing{T}, M::MatElem{T}, charpoly_only::Bool = false) where {T <: RingElement}

Return the minimal polynomial $p$ of the square matrix $M$. If a polynomial ring $S$ over the same base ring as $Y$ is supplied, the resulting polynomial is an element of it.

Examples

julia> R = GF(13)
Finite field F_13

julia> S, y = polynomial_ring(R, :y)
(Univariate polynomial ring in y over R, y)

julia> M = R[7 6 1;
             7 7 5;
             8 12 5]
[7    6   1]
[7    7   5]
[8   12   5]

julia> A = minpoly(S, M)
y^2 + 10*y

julia> A = minpoly(M)
x^2 + 10*x

minpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem

The minimal polynomial of $a$.

minpoly(S::Ring, M::MatRingElem{T}, charpoly_only::Bool = false) where {T <: RingElement}

Return the minimal polynomial $p$ of the matrix $M$. The polynomial ring $S$ of the resulting polynomial must be supplied and the matrix must be square.

charpoly(Y::MatrixElem{T}) where {T <: RingElement}
charpoly(S::PolyRing{T}, Y::MatrixElem{T}) where {T <: RingElement}

Return the characteristic polynomial $p$ of the square matrix $Y$. If a polynomial ring $S$ over the same base ring as $Y$ is supplied, the resulting polynomial is an element of it.

Examples

julia> R, = residue_ring(ZZ, 7);

julia> S = matrix_space(R, 4, 4)
Matrix space of 4 rows and 4 columns
  over residue ring of integers modulo 7

julia> T, y = polynomial_ring(R, :y)
(Univariate polynomial ring in y over R, y)

julia> M = S([R(1) R(2) R(4) R(3); R(2) R(5) R(1) R(0);
              R(6) R(1) R(3) R(2); R(1) R(1) R(3) R(5)])
[1   2   4   3]
[2   5   1   0]
[6   1   3   2]
[1   1   3   5]

julia> A = charpoly(T, M)
y^4 + 2*y^2 + 6*y + 2

julia> A = charpoly(M)
x^4 + 2*x^2 + 6*x + 2

charpoly(a::AbsNumFieldOrderElem) -> ZZPolyRingElem
charpoly(a::AbsNumFieldOrderElem, ZZ) -> ZZPolyRingElem

The characteristic polynomial of $a$.

factor(a::AbsSimpleNumFieldOrderElem) -> Fac{AbsSimpleNumFieldOrderElem}

Computes a factorization of $a$ into irreducible elements. The return value is a factorization fac, which satisfies a = unit(fac) * prod(p^e for (p, e) in fac).

The function requires that $a$ is non-zero and that all prime ideals containing $a$ are principal, which is for example satisfied if class group of the order of $a$ is trivial.

denominator(a::NumFieldElem, O::AbsSimpleNumFieldOrder) -> ZZRingElem

Returns the smallest positive integer $k$ such that $k \cdot a$ is contained in $\mathcal O$.

discriminant(B::Vector{NumFieldOrderElem})

Returns the discriminant of the family $B$ of algebraic numbers, i.e. $det((tr(B[i]*B[j]))_{i, j})^2$.

discriminant(E::EllipticCurve) -> FieldElem

Return the discriminant of $E$.

discriminant(C::HypellCrv{T}) -> T

Compute the discriminant of $C$.

discriminant(O::AlgssRelOrd)

Returns the discriminant of $O$.

discriminant(g::Vector)

Compute the product of all differences of distinct elements in the array.