R = degreesRing 1 t = R_0 h = 1-t^2-2*t^5-t^6+2*t^7+3*t^8-2*t^10 assert( h % (1-t) == 0 ) assert( (h // (1-t)) * (1-t) == h ) debug Core S = QQ[x,y]/(x^2-5) assert ( 1//x * x == 1 ) raw (1_S) // (raw x) S = QQ[x]/(x^2-5) assert ( 1//x * x == 1 ) raw (1_S) // (raw x) use ambient S x rawExtendedGCD(raw x, raw(x^2-5)) R = QQ[x] gcdCoefficients(x,x^2-5) rawExtendedGCD(raw x, raw(x^2-5)) -- this never could have worked: -- R = ZZ[x] -- gcdCoefficients(x,x^2-5) -- rawExtendedGCD(raw x, raw(x^2-5)) R = QQ[x,y]/(x^2-5) 1//x R = QQ[x]/(x^2+1)[y] r = 1_R % (x*1_R) q = 1_R // (x*1_R) assert(1_R - x*q - r == 0) 1_R % gb ideal (x*1_R) remquottest = (f,g) -> ( r = f % g; q = f // g; f - q*g - r) S = ZZ[x,y,z]/(3*x^2-y-1) assert(remquottest(1_S, x^2) == 0) assert(remquottest(y^3, x+1) == 0) (x+1)*q T = ZZ[a]/(a^3-a-1) A = T[x,y,z]/(x*a-1) assert(remquottest(1,x) == 0) assert(remquottest(1,x*y) == 0) assert(remquottest(x*y,a+1) == 0) B = GF(8,Variable=>w)[symbol x]/(x^3-w-1) assert(remquottest(x+w, x^23) == 0)

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