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I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it: ZZ.ideal(5).reduce(17) ZZ.ideal(5).reduce(5) ZZ['x'].ideal(x).reduce(x^5+3) ZZ['x'].ideal(x) 2 0 x^5 + 3 Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring

I had expected to obtain remainder of dividing x^n + 3 /n = 3, not x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/<poly_1>/<poly_2>, the first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )

I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it: ZZ.ideal(5).reduce(17) ZZ.ideal(5).reduce(5) ZZ['x'].ideal(x).reduce(x^5+3) ZZ['x'].ideal(x) 2 0 x^5 + 3 Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring

I had expected to obtain remainder of dividing x^n + 3 /n = 3, not x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/<poly_1>/<poly_2>, ZZ['x']/(poly_1)/(poly_2), the first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )

I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it:
ZZ.ideal(5).reduce(17)
ZZ.ideal(5).reduce(5)
ZZ['x'].ideal(x).reduce(x^5+3)
ZZ['x'].ideal(x)
2
0
x^5 + 3
Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring

I had expected to obtain remainder of dividing x^n + 3 /n = 3, not x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/(poly_1)/(poly_2), the first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )

I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it:
ZZ.ideal(5).reduce(17)
ZZ.ideal(5).reduce(5)
ZZ['x'].ideal(x).reduce(x^5+3)
ZZ['x'].ideal(x)
2
0
x^5 + 3
Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring

I had expected to obtain remainder of dividing x^n + 3 /n = 3, not x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/(poly_1)/(poly_2), the first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )

I am working with a quotient of a quotient of a polynomial ring and I want to make sure the reduction is done correctly. I was able to reduce my problem (hopefully) to the simplest version of it:
ZZ.ideal(5).reduce(17) R.<x>=PolynomialRing(QQ)
ZZ.ideal(5).reduce(5) R.ideal(x^4).reduce(x^8+1)
ZZ['x'].ideal(x).reduce(x^5+3) R.<x>=PolynomialRing(ZZ)
ZZ['x'].ideal(x)
2
0
x^5 R.ideal(x^4).reduce(x^8+1)

1 x^8 + 3
Principal ideal (x) of Univariate Polynomial Ring in x over Integer Ring 1

Why am I had expected to obtain remainder of dividing x^n + 3 /n = 3, not x^5 + 3. What did I do wrong here? How can I ensure I am only working with elements of ZZ['x'] modulus a polynomial of my choice ? (What I actually have is ZZ['x']/(poly_1)/(poly_2), getting the first modular operation works fine but I cannot manage to reduce 2 * poly_2, x * poly_2 to 0 )

EDIT: It works when using QQ as a field, but not when using ZZ. Why?
R.<x>=PolynomialRing(QQ)
R.ideal(R.gen()).reduce(2 * R.gen()+3) result 1 in both cases?