Opened 4 years ago

Closed 4 years ago

# Incorrect division

Reported by: Owned by: pkoprowski major sage-duplicate/invalid/wontfix number fields euclidean division, number field N/A

### Description

Sage incorrectly perform division in (euclidean) rings of algebraic integers. Take:

K.<theta> = QuadraticField(-3)
R = K.OK()


Then $R = \mathbb{Z}\bigl[\frac{1+\sqrt{-3}}{2}\bigr]$ is an Euclidean ring, hence admits division with remainder. Let us take two elements of $R$:

a = R(508*theta); print "a =", a
b = R(1-33*theta); print "b =", b


and divide them:

r = a.mod(b); print "r =", r


The result is $r = 508*\sqrt{-3} = a$ and this is purely wrong as the norms are

print "N(r) =", r.norm()
print "N(b) =", b.norm()


$N(r) = 774192$ and $N(b) = 3268$.

On the other hand, Magma performs the division properly

K<theta> := QuadraticField(-3);
R := RingOfIntegers(K);
a := R!(508*theta); print "a =", a;
b := R!(1-33*theta); print "b =", b;
print "r = ", a mod b;


and returns r = 26*$.2 + 2. Here$.2 is the generator of $R$.

### comment:1 Changed 4 years ago by jdemeyer

Looks like a duplicate of #23971.

### comment:2 Changed 4 years ago by jdemeyer

• Milestone changed from sage-8.1 to sage-duplicate/invalid/wontfix
• Resolution set to duplicate
• Status changed from new to closed
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