Include the fq_default type

setup.py

@@ -105,6 +105,8 @@
     ("flint.types.fmpq_mpoly", ["src/flint/types/fmpq_mpoly.pyx"]),
     ("flint.types.fmpz_mpoly_q", ["src/flint/types/fmpz_mpoly_q.pyx"]),
 
+    ("flint.types.fq_default", ["src/flint/types/fq_default.pyx"]),
+
     ("flint.types.arf", ["src/flint/types/arf.pyx"]),
     ("flint.types.arb", ["src/flint/types/arb.pyx"]),
     ("flint.types.arb_poly", ["src/flint/types/arb_poly.pyx"]),

src/flint/__init__.py

@@ -24,6 +24,8 @@
 
 from .types.fmpq_mpoly import fmpq_mpoly_ctx, fmpq_mpoly, fmpq_mpoly_vec
 
+from .types.fq_default import *
+
 from .types.arf import *
 from .types.arb import *
 from .types.arb_poly import *

src/flint/flint_base/flint_base.pyx

@@ -30,7 +30,127 @@ cdef class flint_elem:
 
 
 cdef class flint_scalar(flint_elem):
-    pass
+    # =================================================
+    # These are the functions a new class should define
+    # assumes that addition and multiplication are
+    # commutative
+    # =================================================
+    def is_zero(self):
+        return False
+
+    def _any_as_self(self):
+        return NotImplemented
+
+    def _neg_(self):
+        return NotImplemented
+
+    def _add_(self, other):
+        return NotImplemented
+
+    def _sub_(self, other, swap=False):
+        return NotImplemented
+
+    def _mul_(self, other):
+        return NotImplemented
+
+    def _div_(self, other, swap=False):
+        return NotImplemented
+
+    def _floordiv_(self, other, swap=False):
+        return NotImplemented
+
+    def _invert_(self):
+        return NotImplemented
+
+    # =================================================
+    # Generic arithmetic using the above functions
+    # =================================================
+
+    def __pos__(self):
+        return self
+
+    def __neg__(self):
+        return self._neg_()
+
+    def __add__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._add_(other)
+
+    def __radd__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._add_(other)
+
+    def __sub__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._sub_(other)
+
+    def __rsub__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._sub_(other, swap=True)
+
+    def __mul__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._mul_(other)
+
+    def __rmul__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._mul_(other)
+
+    def __truediv__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+
+        if other.is_zero():
+            raise ZeroDivisionError
+
+        return self._div_(self, other)
+
+    def __rtruediv__(self, other):
+        if self.is_zero():
+            raise ZeroDivisionError
+
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._div_(other, swap=True)
+
+    def __floordiv__(self, other):
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+
+        if other.is_zero():
+            raise ZeroDivisionError
+
+        return self._floordiv_(self, other)
+
+    def __rfloordiv__(self, other):
+        if self.is_zero():
+            raise ZeroDivisionError
+
+        other = self._any_as_self(other)
+        if other is NotImplemented:
+            return NotImplemented
+        return self._floordiv_(other, swap=True)
+
+    def __invert__(self):
+        if self.is_zero():
+            raise ZeroDivisionError
+        return self._invert_()
+
 
 
 cdef class flint_poly(flint_elem):
@@ -55,7 +175,7 @@ cdef class flint_poly(flint_elem):
         """
         return list(self)
 
-    def str(self, bint ascending=False, *args, **kwargs):
+    def str(self, bint ascending=False, var="x", *args, **kwargs):
         """
         Convert to a human-readable string (generic implementation for
         all polynomial types).
@@ -80,20 +200,20 @@ cdef class flint_poly(flint_elem):
                     s.append("%s" % c)
                 elif i == 1:
                     if c == "1":
-                        s.append("x")
+                        s.append(var)
                     else:
-                        s.append("%s*x" % c)
+                        s.append(f"{c}*{var}")
                 else:
                     if c == "1":
-                        s.append("x^%s" % i)
+                        s.append(f"{var}^{i}")
                     else:
-                        s.append("%s*x^%s" % (c, i))
+                        s.append(f"{c}*{var}^{i}")
         return " + ".join(s)
 
     def roots(self):
         """
         Computes all the roots in the base ring of the polynomial.
-        Returns a list of all pairs (*v*, *m*) where *v* is the 
+        Returns a list of all pairs (*v*, *m*) where *v* is the
         integer root and *m* is the multiplicity of the root.
 
         To compute complex roots of a polynomial, instead use

src/flint/flintlib/fq.pxd

@@ -0,0 +1,109 @@
+from flint.flintlib.flint cimport flint_bitcnt_t, fmpz_struct, slong, flint_rand_t, ulong
+from flint.flintlib.fmpz cimport fmpz_t, fmpz_struct
+from flint.flintlib.fmpz_mod cimport fmpz_mod_ctx_t
+from flint.flintlib.fmpz_mod_mat cimport fmpz_mod_mat_t
+from flint.flintlib.fmpz_poly cimport fmpz_poly_t, fmpz_poly_struct
+from flint.flintlib.fmpz_mod_poly cimport fmpz_mod_poly_t, fmpz_mod_poly_struct
+
+cdef extern from "flint/fq.h":
+
+    ctypedef fmpz_poly_t fq_t
+    ctypedef fmpz_poly_struct fq_struct
+
+    ctypedef struct fq_ctx_struct:
+        fmpz_mod_ctx_t ctxp
+
+        int sparse_modulus
+        int is_conway # whether field was initialized with the Flint Conway tables  (assures primitivity)
+
+        fmpz_struct * a
+        slong * j
+        slong len
+
+        fmpz_mod_poly_t modulus
+        fmpz_mod_poly_t inv
+
+        char * var
+
+    ctypedef fq_ctx_struct fq_ctx_t[1]
+
+    void fq_ctx_init(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
+    int _fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
+    void fq_ctx_init_conway(fq_ctx_t ctx, const fmpz_t p, slong d, const char *var)
+    void fq_ctx_init_modulus(fq_ctx_t ctx, const fmpz_mod_poly_t modulus, const fmpz_mod_ctx_t ctxp, const char *var)
+    void fq_ctx_clear(fq_ctx_t ctx)
+    const fmpz_mod_poly_struct* fq_ctx_modulus(const fq_ctx_t ctx)
+    long fq_ctx_degree(const fq_ctx_t ctx)
+    fmpz_struct * fq_ctx_prime(const fq_ctx_t ctx)
+    void fq_ctx_order(fmpz_t f, const fq_ctx_t ctx)
+    # int fq_ctx_fprint(FILE * file, const fq_ctx_t ctx)
+    void fq_ctx_print(const fq_ctx_t ctx)
+    void fq_ctx_randtest(fq_ctx_t ctx)
+    void fq_ctx_randtest_reducible(fq_ctx_t ctx)
+    void fq_init(fq_t rop, const fq_ctx_t ctx)
+    void fq_init2(fq_t rop, const fq_ctx_t ctx)
+    void fq_clear(fq_t rop, const fq_ctx_t ctx)
+    void _fq_sparse_reduce(fmpz_struct *R, slong lenR, const fq_ctx_t ctx)
+    void _fq_dense_reduce(fmpz_struct *R, slong lenR, const fq_ctx_t ctx)
+    void _fq_reduce(fmpz_struct *r, slong lenR, const fq_ctx_t ctx)
+    void fq_reduce(fq_t rop, const fq_ctx_t ctx)
+    void fq_add(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
+    void fq_sub(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
+    void fq_sub_one(fq_t rop, const fq_t op1, const fq_ctx_t ctx)
+    void fq_neg(fq_t rop, const fq_t op, const fq_ctx_t ctx)
+    void fq_mul(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
+    void fq_mul_fmpz(fq_t rop, const fq_t op, const fmpz_t x, const fq_ctx_t ctx)
+    void fq_mul_si(fq_t rop, const fq_t op, slong x, const fq_ctx_t ctx)
+    void fq_mul_ui(fq_t rop, const fq_t op, ulong x, const fq_ctx_t ctx)
+    void fq_sqr(fq_t rop, const fq_t op, const fq_ctx_t ctx)
+    void fq_div(fq_t rop, const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
+    void _fq_inv(fmpz_struct *rop, const fmpz_struct *op, slong len, const fq_ctx_t ctx)
+    void fq_inv(fq_t rop, const fq_t op, const fq_ctx_t ctx)
+    void fq_gcdinv(fq_t f, fq_t inv, const fq_t op, const fq_ctx_t ctx)
+    void _fq_pow(fmpz_struct *rop, const fmpz_struct *op, slong len, const fmpz_t e, const fq_ctx_t ctx)
+    void fq_pow(fq_t rop, const fq_t op, const fmpz_t e, const fq_ctx_t ctx)
+    void fq_pow_ui(fq_t rop, const fq_t op, const ulong e, const fq_ctx_t ctx)
+    int fq_sqrt(fq_t rop, const fq_t op1, const fq_ctx_t ctx)
+    void fq_pth_root(fq_t rop, const fq_t op1, const fq_ctx_t ctx)
+    int fq_is_square(const fq_t op, const fq_ctx_t ctx)
+    # int fq_fprint_pretty(FILE *file, const fq_t op, const fq_ctx_t ctx)
+    int fq_print_pretty(const fq_t op, const fq_ctx_t ctx)
+    # void fq_fprint(FILE * file, const fq_t op, const fq_ctx_t ctx)
+    void fq_print(const fq_t op, const fq_ctx_t ctx)
+    char * fq_get_str(const fq_t op, const fq_ctx_t ctx)
+    char * fq_get_str_pretty(const fq_t op, const fq_ctx_t ctx)
+    void fq_randtest(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
+    void fq_randtest_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
+    void fq_randtest_dense(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
+    void fq_rand(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
+    void fq_rand_not_zero(fq_t rop, flint_rand_t state, const fq_ctx_t ctx)
+    void fq_set(fq_t rop, const fq_t op, const fq_ctx_t ctx)
+    void fq_set_si(fq_t rop, const slong x, const fq_ctx_t ctx)
+    void fq_set_ui(fq_t rop, const ulong x, const fq_ctx_t ctx)
+    void fq_set_fmpz(fq_t rop, const fmpz_t x, const fq_ctx_t ctx)
+    void fq_swap(fq_t op1, fq_t op2, const fq_ctx_t ctx)
+    void fq_zero(fq_t rop, const fq_ctx_t ctx)
+    void fq_one(fq_t rop, const fq_ctx_t ctx)
+    void fq_gen(fq_t rop, const fq_ctx_t ctx)
+    int fq_get_fmpz(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)
+    void fq_get_fmpz_poly(fmpz_poly_t a, const fq_t b, const fq_ctx_t ctx)
+    void fq_get_fmpz_mod_poly(fmpz_mod_poly_t a, const fq_t b, const fq_ctx_t ctx)
+    void fq_set_fmpz_poly(fq_t a, const fmpz_poly_t b, const fq_ctx_t ctx)
+    void fq_set_fmpz_mod_poly(fq_t a, const fmpz_mod_poly_t b, const fq_ctx_t ctx)
+    void fq_get_fmpz_mod_mat(fmpz_mod_mat_t col, const fq_t a, const fq_ctx_t ctx)
+    void fq_set_fmpz_mod_mat(fq_t a, const fmpz_mod_mat_t col, const fq_ctx_t ctx)
+    int fq_is_zero(const fq_t op, const fq_ctx_t ctx)
+    int fq_is_one(const fq_t op, const fq_ctx_t ctx)
+    int fq_equal(const fq_t op1, const fq_t op2, const fq_ctx_t ctx)
+    int fq_is_invertible(const fq_t op, const fq_ctx_t ctx)
+    int fq_is_invertible_f(fq_t f, const fq_t op, const fq_ctx_t ctx)
+    void _fq_trace(fmpz_t rop, const fmpz_struct *op, slong len, const fq_ctx_t ctx)
+    void fq_trace(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)
+    void _fq_norm(fmpz_t rop, const fmpz_struct *op, slong len, const fq_ctx_t ctx)
+    void fq_norm(fmpz_t rop, const fq_t op, const fq_ctx_t ctx)
+    void _fq_frobenius(fmpz_struct *rop, const fmpz_struct *op, slong len, slong e, const fq_ctx_t ctx)
+    void fq_frobenius(fq_t rop, const fq_t op, slong e, const fq_ctx_t ctx)
+    int fq_multiplicative_order(fmpz_t ord, const fq_t op, const fq_ctx_t ctx)
+    int fq_is_primitive(const fq_t op, const fq_ctx_t ctx)
+    void fq_bit_pack(fmpz_t f, const fq_t op, flint_bitcnt_t bit_size, const fq_ctx_t ctx)
+    void fq_bit_unpack(fq_t rop, const fmpz_t f, flint_bitcnt_t bit_size, const fq_ctx_t ctx)