Mod

Function: Mod
Section: conversions
C-Name: gmodulo
Prototype: GG
Help: Mod(a,b): create 'a modulo b'.
Description:
 (small, small):gen         gmodulss($1, $2)
 (small, gen):gen           gmodulsg($1, $2)
 (gen, gen):gen             gmodulo($1, $2)
Doc: in its basic form, create an intmod or a polmod $(a \mod b)$; $b$ must
 be an integer or a polynomial. We then obtain a \typ{INTMOD} and a
 \typ{POLMOD} respectively:
 \bprog
 ? t = Mod(2,17); t^8
 %1 = Mod(1, 17)
 ? t = Mod(x,x^2+1); t^2
 %2 = Mod(-1, x^2+1)
 @eprog\noindent If $a \% b$ makes sense and yields a result of the
 appropriate type (\typ{INT} or scalar/\typ{POL}), the operation succeeds as
 well:
 \bprog
 ? Mod(1/2, 5)
 %3 = Mod(3, 5)
 ? Mod(7 + O(3^6), 3)
 %4 = Mod(1, 3)
 ? Mod(Mod(1,12), 9)
 %5 = Mod(1, 3)
 ? Mod(1/x, x^2+1)
 %6 = Mod(-x, x^2+1)
 ? Mod(exp(x), x^4)
 %7 = Mod(1/6*x^3 + 1/2*x^2 + x + 1, x^4)
 @eprog
 If $a$ is a complex object, ``base change'' it to $\Z/b\Z$ or $K[x]/(b)$,
 which is equivalent to, but faster than, multiplying it by \kbd{Mod(1,b)}:
 \bprog
 ? Mod([1,2;3,4], 2)
 %8 =
 [Mod(1, 2) Mod(0, 2)]

 [Mod(1, 2) Mod(0, 2)]
 ? Mod(3*x+5, 2)
 %9 = Mod(1, 2)*x + Mod(1, 2)
 ? Mod(x^2 + y*x + y^3, y^2+1)
 %10 = Mod(1, y^2 + 1)*x^2 + Mod(y, y^2 + 1)*x + Mod(-y, y^2 + 1)
 @eprog

 This function is not the same as $x$ \kbd{\%} $y$, the result of which
 has no knowledge of the intended modulus $y$. Compare
 \bprog
 ? x = 4 % 5; x + 1
 %11 = 5
 ? x = Mod(4,5); x + 1
 %12 = Mod(0,5)
 @eprog Note that such ``modular'' objects can be lifted via \tet{lift} or
 \tet{centerlift}. The modulus of a \typ{INTMOD} or \typ{POLMOD} $z$ can
 be recovered via \kbd{$z$.mod}.