Overview Giac provides comprehensive support for algebraic extensions of Q (the rationals), allowing computations in fields defined by minimal polynomials. The API handles root representations, extension field arithmetic, and conversions between symbolic and algebraic forms.
Algebraic Extension Construction algebraic_EXTension() gen algebraic_EXTension ( const gen & a , const gen & v );
Creates an algebraic extension element from a representation a and a variable v.
rootof() gen rootof ( const gen & e , GIAC_CONTEXT );
Creates a root-of representation for solving polynomial equations.
gen symb_rootof ( const gen & p , const gen & pmin , GIAC_CONTEXT );
Symbolic form: rootof(p, pmin) represents a root of minimal polynomial pmin selected by p.
C++ Example
Python Example
// Create rootof(x^2-2) - represents sqrt(2) gen pmin = makevecteur ( 1 , 0 , - 2 ); // x^2 - 2 gen root = rootof (pmin, context);
ext_reduce() gen ext_reduce ( const gen & a , const gen & v ); gen ext_reduce ( const gen & e ); void clean_ext_reduce ( vecteur & v ); void clean_ext_reduce ( gen & g );
Reduces expressions in algebraic extensions to canonical form.
Extension Field Arithmetic ext_add() gen ext_add ( const gen & a , const gen & b , GIAC_CONTEXT );
Adds two elements in an algebraic extension field.
ext_sub() gen ext_sub ( const gen & a , const gen & b , GIAC_CONTEXT );
Subtracts two elements in an algebraic extension field.
ext_mul() gen ext_mul ( const gen & a , const gen & b , GIAC_CONTEXT );
Multiplies two elements in an algebraic extension field.
inv_EXT() gen inv_EXT ( const gen & a );
Computes the multiplicative inverse in an algebraic extension field.
gen sqrt2 = rootof ( makevecteur ( 1 , 0 , - 2 ), context); gen sqrt3 = rootof ( makevecteur ( 1 , 0 , - 3 ), context); // Arithmetic in extension field gen sum = ext_add (sqrt2, sqrt3, context); gen product = ext_mul (sqrt2, sqrt3, context); // sqrt(6) gen inverse = inv_EXT (sqrt2); // sqrt(2)/2
Common Extension Field common_EXT() gen common_EXT ( gen & a , gen & b , const vecteur * l , GIAC_CONTEXT );
Finds a common extension field for two algebraic elements and converts them to that field.
First element (modified to be in common field)
Second element (modified to be in common field)
Optional list of variables
The minimal polynomial of the common extension
common_minimal_POLY() gen common_minimal_POLY ( const gen & ga , const gen & gb , gen & a , gen & b , int & k , const vecteur * l , GIAC_CONTEXT );
Computes the common minimal polynomial and converts elements.
Minimal Polynomial Operations min_pol() vecteur min_pol ( gen & a );
Extracts the minimal polynomial from an algebraic extension element.
minimal_polynomial() gen minimal_polynomial ( const gen & pp , bool minonly , GIAC_CONTEXT );
Computes the minimal polynomial of an algebraic expression.
Root Selection and Evaluation select_root() gen select_root ( const vecteur & v , GIAC_CONTEXT );
Selects a specific root from a list of roots of a polynomial.
gen in_select_root ( const vecteur & a , bool reel , GIAC_CONTEXT , double eps = 1 e- 14 );
Selects a root with optional constraint to real roots.
horner_rootof() gen horner_rootof ( const vecteur & p , const gen & g , GIAC_CONTEXT );
Evaluates a polynomial at a root-of expression using Horner’s method.
has_rootof_value() bool has_rootof_value ( const gen & Pmin , gen & value , GIAC_CONTEXT );
Checks if a root-of expression has a known numeric value.
Numerical Approximation alg_evalf() gen alg_evalf ( const gen & a , const gen & b , const gen & c , GIAC_CONTEXT );
Evaluates an algebraic extension element to floating-point.
approx_rootof() gen approx_rootof ( const gen & e , GIAC_CONTEXT );
Computes numerical approximation of a root-of expression.
gen root = rootof ( makevecteur ( 1 , 0 , - 2 ), context); gen approx = approx_rootof (root, context); // approx ≈ 1.41421356...
Root Caching proot_cached() / proot_cache() bool proot_cached ( const vecteur & v , double eps , vecteur & res ); bool proot_cache ( const vecteur & v , double eps , const vecteur & res );
Cache system for polynomial roots to avoid recomputation.
galoisconj_cached() / galoisconj_cache() bool galoisconj_cached ( const vecteur & v , vecteur & res ); bool galoisconj_cache ( const vecteur & v , const vecteur & res );
Cache system for Galois conjugates.
Galois Theory galoisconj() vecteur galoisconj ( const vecteur & v , GIAC_CONTEXT );
Computes all Galois conjugates of an algebraic element.
conj_in_nf() bool conj_in_nf ( const vecteur & w , gen & g , GIAC_CONTEXT );
Checks if conjugates are in a number field.
Variable Management lvar() vecteur lvar ( const gen & e ); void lvar ( const gen & e , vecteur & l );
Extracts list of variables from an expression.
alg_lvar() void alg_lvar ( const gen & e , matrice & m ); matrice alg_lvar ( const gen & e );
Extracts variables organized by algebraic extension levels.
Extension Format : The result is a matrix where each row represents variables at one extension level:
First row: base variables (not in extensions)
Subsequent rows: variables in each extension level
Conversion Functions sym2r() / r2sym() fraction sym2r ( const gen & e , const vecteur & l , GIAC_CONTEXT ); bool sym2r ( const gen & e , const vecteur & l , int l_size , gen & num , gen & den , GIAC_CONTEXT );
Converts symbolic expressions to internal polynomial representation.
gen r2sym ( const gen & p , const vecteur & l , GIAC_CONTEXT ); gen r2e ( const gen & p , const vecteur & l , GIAC_CONTEXT );
Converts internal polynomial form back to symbolic expressions.
vecteur vars = makevecteur (x, y); gen expr = pow (x, 2 ) + 2 * x * y + pow (y, 2 ); // Convert to internal form fraction f = sym2r (expr, vars, context); // Convert back to symbolic gen result = r2sym (f, vars, context);
algnum_normal() bool algnum_normal ( gen & e , GIAC_CONTEXT );
Simplifies expressions in real algebraic extensions of Q. Requires Gröbner basis implementation.
algnum_rref() bool algnum_rref ( const matrice & a , matrice & res , vecteur & pivots , gen & det , int redtype , GIAC_CONTEXT );
Row reduction for matrices with algebraic number coefficients.
0: rref with full reduction
1: rref upper triangular
2: compute determinant only
3: LU decomposition
Utility Functions is_sqrt() bool is_sqrt ( const gen & a , gen & arg );
Tests if an expression is a square root and extracts the argument.
islesscomplex() bool islesscomplex ( const gen & a , const gen & b );
Compares complexity of two expressions for ordering.
is_known_rootof() bool is_known_rootof ( const vecteur & v , const vecteur & lv , gen & symroot , GIAC_CONTEXT );
Checks if a root is in the known root database.
Global Maps rootmap & symbolic_rootof_list (); rootmap & proot_list (); rootmap & galoisconj_list ();
Access global caches for symbolic roots, polynomial roots, and Galois conjugates.
Example: Working with Q(√2, √3) #include "alg_ext.h" void example_extension_field () { context ctx; // Create sqrt(2) vecteur pmin2 = makevecteur ( 1 , 0 , - 2 ); // x^2 - 2 gen sqrt2 = rootof (pmin2, & ctx); // Create sqrt(3) vecteur pmin3 = makevecteur ( 1 , 0 , - 3 ); // x^2 - 3 gen sqrt3 = rootof (pmin3, & ctx); // Find common extension Q(sqrt(2), sqrt(3)) = Q(sqrt(6)) gen a = sqrt2, b = sqrt3; gen common_poly = common_EXT (a, b, nullptr , & ctx); // Perform arithmetic in the extension field gen sum = ext_add (sqrt2, sqrt3, & ctx); gen prod = ext_mul (sqrt2, sqrt3, & ctx); // sqrt(6) // Reduce to normal form gen reduced = ext_reduce (sum, & ctx); // Get numerical approximation gen approx_sum = alg_evalf (sum, 0 , 0 , & ctx); // approx_sum ≈ 3.146... }
See Also
Polynomial API Core polynomial operations
Equation Solving Solve equations with algebraic extensions
