Use number-field element.

nf_elem_init a nf

Initialise a number field element to belong to the given number field code{nf}. The element is set to zero.

nf_elem_clear a nf

Clear resources allocated by the given number field element in the given number field.

nf_elem_randtest a state bits nf

Generate a random number field element \(a\) in the number field code{nf} whose coefficients have up to the given number of bits.

nf_elem_canonicalise a nf

Canonicalise a number field element, i.e. reduce numerator and denominator to lowest terms. If the numerator is \(0\), set the denominator to \(1\).

_nf_elem_reduce a nf

Reduce a number field element modulo the defining polynomial. This is used with functions such as code{nf_elem_mul_red} which allow reduction to be delayed. Does not canonicalise.

nf_elem_reduce a nf

Reduce a number field element modulo the defining polynomial. This is used with functions such as code{nf_elem_mul_red} which allow reduction to be delayed.

_nf_elem_invertible_check a nf

Whilst the defining polynomial for a number field should by definition be irreducible, it is not enforced. Thus in test code, it is convenient to be able to check that a given number field element is invertible modulo the defining polynomial of the number field. This function does precisely this.

If \(a\) is invertible modulo the defining polynomial of code{nf} the value \(1\) is returned, otherwise \(0\) is returned.

The function is only intended to be used in test code.

nf_elem_set_fmpz_mat_row b M i den nf

Set \(b\) to the element specified by row \(i\) of the matrix \(M\) and with the given denominator \(d\). Column \(0\) of the matrix corresponds to the constant coefficient of the number field element.

nf_elem_get_fmpz_mat_row M i den b nf

Set the row \(i\) of the matrix \(M\) to the coefficients of the numerator of the element \(b\) and \(d\) to the denominator of \(b\). Column \(0\) of the matrix corresponds to the constant coefficient of the number field element.

nf_elem_set_fmpq_poly a pol nf

Set \(a\) to the element corresponding to the polynomial code{pol}.

nf_elem_get_fmpq_poly pol a nf

Set code{pol} to a polynomial corresponding to \(a\), reduced modulo the defining polynomial of code{nf}.

nf_elem_get_nmod_poly_den pol a nf den

Set code{pol} to the reduction of the polynomial corresponding to the numerator of \(a\). If code{den == 1}, the result is multiplied by the inverse of the denominator of \(a\). In this case it is assumed that the reduction of the denominator of \(a\) is invertible.

nf_elem_get_nmod_poly pol a nf

Set code{pol} to the reduction of the polynomial corresponding to the numerator of \(a\). The result is multiplied by the inverse of the denominator of \(a\). It is assumed that the reduction of the denominator of \(a\) is invertible.

nf_elem_get_fmpz_mod_poly_den pol a nf den

Set code{pol} to the reduction of the polynomial corresponding to the numerator of \(a\). If code{den == 1}, the result is multiplied by the inverse of the denominator of \(a\). In this case it is assumed that the reduction of the denominator of \(a\) is invertible.

nf_elem_get_fmpz_mod_poly pol a nf

Set code{pol} to the reduction of the polynomial corresponding to the numerator of \(a\). The result is multiplied by the inverse of the denominator of \(a\). It is assumed that the reduction of the denominator of \(a\) is invertible.

nf_elem_set_den b d nf

Set the denominator of the code{nf_elem_t b} to the given integer \(d\). Assumes \(d > 0\).

nf_elem_get_den d b nf

Set \(d\) to the denominator of the code{nf_elem_t b}.

_nf_elem_set_coeff_num_fmpz a i d nf

Set the \(i`th coefficient of the denominator of :math:`a\) to the given integer \(d\).

_nf_elem_equal a b nf

Return \(1\) if the given number field elements are equal in the given number field code{nf}. This function does emph{not} assume \(a\) and \(b\) are canonicalised.

nf_elem_equal a b nf

Return \(1\) if the given number field elements are equal in the given number field code{nf}. This function assumes \(a\) and \(b\) emph{are} canonicalised.

nf_elem_is_zero a nf

Return \(1\) if the given number field element is equal to zero, otherwise return \(0\).

nf_elem_is_one a nf

Return \(1\) if the given number field element is equal to one, otherwise return \(0\).

nf_elem_print_pretty a nf var

Print the given number field element to code{stdout} using the null-terminated string code{var} not equal to code{"0"} as the name of the primitive element. foreign import ccall "nf_elem.h nf_elem_print_pretty"

nf_elem_zero a nf

Set the given number field element to zero.

nf_elem_one a nf

Set the given number field element to one.

nf_elem_set a b nf

Set the number field element \(a\) to equal the number field element \(b\), i.e. set \(a = b\).

nf_elem_neg a b nf

Set the number field element \(a\) to minus the number field element \(b\), i.e. set \(a = -b\).

nf_elem_swap a b nf

Efficiently swap the two number field elements \(a\) and \(b\).

nf_elem_mul_gen a b nf

Multiply the element \(b\) with the generator of the number field.

_nf_elem_add r a b nf

Add two elements of a number field code{nf}, i.e. set \(r = a + b\). Canonicalisation is not performed.

nf_elem_add r a b nf

Add two elements of a number field code{nf}, i.e. set \(r = a + b\).

_nf_elem_sub r a b nf

Subtract two elements of a number field code{nf}, i.e. set \(r = a - b\). Canonicalisation is not performed.

nf_elem_sub r a b nf

Subtract two elements of a number field code{nf}, i.e. set \(r = a - b\).

_nf_elem_mul a b c nf

Multiply two elements of a number field code{nf}, i.e. set \(r = a * b\). Does not canonicalise. Aliasing of inputs with output is not supported.

_nf_elem_mul_red a b c nf red

As per code{_nf_elem_mul}, but reduction modulo the defining polynomial of the number field is only carried out if code{red == 1}. Assumes both inputs are reduced.

nf_elem_mul a b c nf

Multiply two elements of a number field code{nf}, i.e. set \(r = a * b\).

nf_elem_mul_red a b c nf red

As per code{nf_elem_mul}, but reduction modulo the defining polynomial of the number field is only carried out if code{red == 1}. Assumes both inputs are reduced.

_nf_elem_inv r a nf

Invert an element of a number field code{nf}, i.e. set \(r = a^{-1}\). Aliasing of the input with the output is not supported.

nf_elem_inv r a nf

Invert an element of a number field code{nf}, i.e. set \(r = a^{-1}\).

_nf_elem_div a b c nf

Set \(a\) to \(b/c\) in the given number field. Aliasing of \(a\) and \(b\) is not permitted.

nf_elem_div a b c nf

Set \(a\) to \(b/c\) in the given number field.

_nf_elem_pow res a e nf

Set code{res} to \(a^e\) using left-to-right binary exponentiation as described in~citep[p.~461]{Knu1997}.

Assumes that \(a \neq 0\) and \(e > 1\). Does not support aliasing.

nf_elem_pow res a e nf

Set code{res} = code{a^e} using the binary exponentiation algorithm. If \(e\) is zero, returns one, so that in particular code{0^0 = 1}.

_nf_elem_norm rnum rden a nf

Set code{{rnum, rden}} to the absolute norm of the given number field element \(a\).

nf_elem_norm res a nf

Set code{res} to the absolute norm of the given number field element \(a\).

nf_elem_norm_div res a nf div nbits

Set code{res} to the absolute norm of the given number field element \(a\), divided by code{div} . Assumes the result to be an integer and having at most code{nbits} bits.

_nf_elem_norm_div rnum rden a nf divisor nbits

Set code{{rnum, rden}} to the absolute norm of the given number field element \(a\), divided by code{div} . Assumes the result to be an integer and having at most code{nbits} bits.

_nf_elem_trace rnum rden a nf

Set code{{rnum, rden}} to the absolute trace of the given number field element \(a\).

nf_elem_trace res a nf

Set code{res} to the absolute trace of the given number field element \(a\).

nf_elem_rep_mat res a nf

Set code{res} to the matrix representing the multiplication with \(a\) with respect to the basis \(1, a, \dotsc, a^{d - 1}\), where \(a\) is the generator of the number field of \(d\) is its degree.

nf_elem_rep_mat_fmpz_mat_den res den a nf

Return a tuple \(M, d\) such that \(M/d\) is the matrix representing the multiplication with \(a\) with respect to the basis \(1, a, \dotsc, a^{d - 1}\), where \(a\) is the generator of the number field of \(d\) is its degree. The integral matrix \(M\) is primitive.

nf_elem_mod_fmpz_den z a mod nf

If code{den == 0}, return an element \(z\) with denominator \(1\), such that the coefficients of \(z - da\) are divisble by code{mod}, where \(d\) is the denominator of \(a\). The coefficients of \(z\) are reduced modulo code{mod}.

If code{den == 1}, return an element \(z\), such that \(z - a\) has denominator \(1\) and the coefficients of \(z - a\) are divisble by code{mod}. The coefficients of \(z\) are reduced modulo code{mod * d}, where \(d\) is the denominator of \(a\).

Reduction takes place with respect to the positive residue system.

nf_elem_smod_fmpz_den z a mod nf

If code{den == 0}, return an element \(z\) with denominator \(1\), such that the coefficients of \(z - da\) are divisble by code{mod}, where \(d\) is the denominator of \(a\). The coefficients of \(z\) are reduced modulo code{mod}.

If code{den == 1}, return an element \(z\), such that \(z - a\) has denominator \(1\) and the coefficients of \(z - a\) are divisble by code{mod}. The coefficients of \(z\) are reduced modulo code{mod * d}, where \(d\) is the denominator of \(a\).

Reduction takes place with respect to the symmetric residue system.

nf_elem_mod_fmpz res a mod nf

Return an element \(z\) such that \(z - a\) has denominator \(1\) and the coefficients of \(z - a\) are divisible by code{mod}. The coefficients of \(z\) are reduced modulo code{mod * d}, where \(d\) is the denominator of \(b\).

Reduction takes place with respect to the positive residue system.

nf_elem_smod_fmpz res a mod nf

Return an element \(z\) such that \(z - a\) has denominator \(1\) and the coefficients of \(z - a\) are divisible by code{mod}. The coefficients of \(z\) are reduced modulo code{mod * d}, where \(d\) is the denominator of \(b\).

Reduction takes place with respect to the symmetric residue system.

nf_elem_coprime_den res a mod nf

Return an element \(z\) such that the denominator of \(z - a\) is coprime to code{mod}.

Reduction takes place with respect to the positive residue system.

nf_elem_coprime_den_signed res a mod nf

Return an element \(z\) such that the denominator of \(z - a\) is coprime to code{mod}.

Reduction takes place with respect to the symmetric residue system.