ca_field.h

Initialises K as the field \mathbb{Q}.

Initialises K as the number field \mathbb{Q}(x), inserting the extension number x into the cache of ctx.

Initialises K as the generic field \mathbb{Q}(a) with the single generator a given by the constant func (for example CA_Pi), the function value f(x) or the function value f(x, y), inserting the extension number into the cache of ctx. The ideal is empty; it is not built by these functions.

Initialises K as a generic field with len generators, which must then be assigned one by one with ca_field_set_ext. The ideal is empty.

Sets the generator with index i of K to the extension number x and updates the hash. Only a pointer is stored, so x must outlive K; extension numbers from the cache of the context do.

Clears K and its ideal. The extension numbers are not cleared, since the field only holds references to them.

Prints a description of K to standard output, of the form QQ(x1, x2) where {x1 = ..., x2 = ...} with ideal {...}, with the generators printed by ca_ext_print and the ideal (if not empty) in the variables x1, x2, ....

Builds the ideal of relations of the generic field K, whose generators have been assigned, in place. Nothing is done for \mathbb{Q} and for number fields. The relations found are always valid, but they need not generate the full ideal of relations between the generators. If CA_OPT_USE_GROEBNER is set and the relations have total degree at most CA_OPT_QQBAR_DEG_LIMIT, the ideal is replaced by its reduced Gröbner basis when Buchberger's algorithm finishes within the limits set by the context options. See Formal fields and the ideal of relations for the relations that are searched for.

Adds relations between error functions among the generators of K that follow from \operatorname{erf}(-x) = -\operatorname{erf}(x), \operatorname{erfc}(x) = 1 - \operatorname{erf}(x) and \operatorname{erfi}(x) = -i \operatorname{erf}(ix), when the arguments are proved to be equal or opposite (or related by a factor i). Called by ca_field_build_ideal.

Adds the relations \Gamma(x + n) = x (x+1) \cdots (x + n - 1) \Gamma(x) between generators \Gamma(x + n) and \Gamma(x) of K whose arguments differ by an integer n with |n| \le 10 (and \Gamma(x) = \Gamma(y) when x = y), provided the factors x + k can be expressed in K. Called by ca_field_build_ideal.

Compares the fields K1 and K2 in a total structural order (number of generators, then the generators with ca_ext_cmp_repr), returning -1, 0 or 1. The order says nothing about the fields as mathematical objects.

Initialises cache as empty, or clears it, freeing all fields in it.

Returns a pointer to the field in cache whose generators are the length extension numbers x (compared as pointers, in the given order), creating it if needed. A newly created field is \mathbb{Q} if length is 0, a number field if it has one algebraic generator, and otherwise a generic field whose ideal is built immediately with ca_field_build_ideal. The extension numbers must belong to the cache of ctx.