FLINT3.7.0-dev
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Module · 12 functions

nf.h

#include <flint/nf.h>

This module defines the type nf_t of a number field K = \mathbf{Q}(\theta) = \mathbf{Q}[x]/(f), given by a defining polynomial f \in \mathbf{Q}[x] of degree d \ge 1, and computes the data that the element arithmetic of nf_elem.h needs: the representation used for elements (special ones for degree 1 and 2), a precomputed inverse of the leading coefficient, the powers x^i \bmod f used for reduction, and the power sums of the roots of f used for traces. The module goes back to the ANTIC library [Har2015]. The generic ring interface for number fields is gr_ctx_init_nf in gr.h; algebraic number fields as subfields of \mathbf{C} are handled by qqbar.h and ca.h.

AlgorithmArithmetic in number fieldsHow nf_elem represents elements of Q[x]/(f) with special cases for degrees 1 and 2 and for Q(i), multiplies with reduction by precomputed powers of the generator, inverts by the extended Euclidean algorithm, and computes norms as resultants and traces from power sums of the roots.

Types#

A number field. The fields are pol (the defining polynomial, an fmpq_poly_t), pinv (a precomputed inverse of the leading coefficient of the numerator of pol, used for division in the non-monic generic case), powers (precomputed reductions of x^{d}, \ldots, x^{2d-2} modulo pol for 3 \le d < NF_POWERS_CUTOFF), traces (the power sums S_k = \sum_i \theta_i^k of the roots \theta_i of pol for 0 \le k < d, as an fmpq_poly_t) and flag.

Bits of the field flag. NF_MONIC is set if pol has integer coefficients and leading coefficient 1, NF_LINEAR for d = 1, NF_QUADRATIC for d = 2, and NF_GAUSSIAN in addition for f = x^2 + 1 exactly; NF_GENERIC is the value 0. Elements of linear and quadratic fields use the representations lnf_elem_t and qnf_elem_t of nf_elem.h.

The bound 30 on the length of the defining polynomial up to which the powers of x modulo f are precomputed for reduction.

Functions#

void nf_init(nf_t nf, const fmpq_poly_t pol)

Initialises nf for the defining polynomial pol, which must have degree at least 1 (otherwise the function throws). The polynomial is not checked to be irreducible: with a reducible polynomial the arithmetic is that of the ring \mathbf{Q}[x]/(f), and the result of inverting a zero divisor is meaningless (_nf_elem_invertible_check tests for this).

More
void nf_init_randtest(nf_t nf, flint_rand_t state, slong len, flint_bitcnt_t bits_in)

Initialises nf with a random defining polynomial of length at most len (that is, degree at most len - 1) with coefficients of at most bits_in bits, for testing. Linear and quadratic fields, monic polynomials and x^2 + 1 are chosen with increased probability. The polynomial need not be irreducible. Throws if len < 2 or bits_in < 1.

void nf_clear(nf_t nf)

Clears nf.

void nf_print(const nf_t nf)

Prints a description of nf, with the defining polynomial in the variable x, to stdout, without a final newline.

References#

  • [Har2015] W. B. Hart. "ANTIC: Algebraic number theory in C". Computeralgebra-Rundbrief: Vol. 56, 2015
Cite this page

The FLINT team, “nf.h (module overview),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/nf/index.html (generated 2026-10-20).