Ask Your Question

Revision history [back]

click to hide/show revision 1
initial version

Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2
a^2
sage: 2*a
2*a
sage: S(2)+S(2)
4

When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0

But now look at this:

sage: S(2)+S(3)
5

How can I solve this problem?

Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2
a^2    # the output should be 0
a^2
sage: 2*a
2*a    # the output should be 0
2*a
sage: S(2)+S(2)
S(2)+S(2)    # the output should be 0
4

When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0

But now look at this:

sage: S(2)+S(3)
S(2)+S(3)    # the output should be 1
5

How can I solve this problem?

Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2    # the output should be 0
a^2
sage: 2*a    # the output should be 0
2*a
sage: S(2)+S(2)    # the output should be 0
4

When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$, Z[x,y]/\langle4,2x,x^2,y\rangle$ instead, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0

But now look at this:

sage: S(2)+S(3)    # the output should be 1
5

How can I solve this problem?

Quotient of polynomial ring over integers not working

I have a problem. I want Sage to calculate in $\mathbb Z[x]/\langle4,2x,x^2\rangle$ but none of the relations are being calculated/recognized properly:

sage: R.<x> = ZZ[]
sage: I = R.ideal(4,2*x,x^2)
sage: S.<a> = R.quotient(I)
sage: a^2    # the output should be 0
a^2
sage: 2*a    # the output should be 0
2*a
sage: S(2)+S(2)    # the output should be 0
4

When I introduce a superfluous variable $y$ and consider $\mathbb Z[x,y]/\langle4,2x,x^2,y\rangle$ instead, which is practically the same ring (i.e. isomorphic), then it seems to work:

sage: R.<x,y> = ZZ[]
sage: I = R.ideal(4,2*x,x^2,y)
sage: S.<a,b> = R.quotient(I)
sage: a^2
0
sage: 2*a
0
sage: S(2)+S(2)
0

But now look at this:

sage: S(2)+S(3)    # the output should be 1
5

How can I solve this problem?