# Trying to understand a Univariate Quotient Polynomial Ring over Finite Field

This might be a silly question and I am sorry for my lack of knowledge in abstract algebra.

I am trying to understand a Univariate Quotient Polynomial Ring over Finite Field but I want to make sure I am correct. So, if we have a polynomial, for example: $p = c_1 + c_2x^1 + c_3x^2 + \dots \ c_nx^{n-1}$ where $c_{i} \in \mathbb{Z}$.

Now to bring this polynomial into Quotient Polynomial Ring over Finite Field, for example: $r = x^{1024} + 1$ as a irreducible polynomial and some prime $q$ as a modulus.

First we need to find a quotient of division of $p / r$ and then for every coefficient in $p$ we need to $\bmod q$. Am I right? or completely off track?

I just want to know how SageMath does something like this:

```
modulus = 12289 # some prime
R.<X> = PolynomialRing(GF(modulus))
Y.<x> = R.quotient(X^(1024) + 1)
```

The reason I am asking this question is because I wrote something like this to simulate above SageMath code but it returns false:

```
import random
from itertools import izip
from math import fabs
dimension = 25 # degree of polynomials
modulus = 40961 # modulus
# ring = x^dimension + 1
ring = [1] + [0 for _ in range(dimension - 1)] + [1]
# Quotient polynomial ring
R.<X> = PolynomialRing(GF(modulus)) # Gaussian field of integers
Y.<x> = R.quotient(X^(dimension) + 1) # Cyclotomic field
def division(N, D):
dD = degree(D)
dN = degree(N)
if dD < 0: raise ZeroDivisionError
if dN >= dD:
q = [0] * dN
while dN >= dD:
d = [0] * (dN - dD) + D
mult = q[dN - dD] = N[-1] / d[-1]
d = [coeff * mult for coeff in d]
N = [abs(coeffN - coeffd) for coeffN, coeffd in izip(N, d)]
dN = degree(N)
r = N
else:
q = [0]
r = N
return (q, r)
def degree(poly):
while poly and poly[-1] == 0:
poly.pop() # normalize
return len(poly) - 1
def gen():
return [random.randint(100, 200) for _ in range(2^11)]
def ring_correct(poly, irreduciblePoly, modulus):
poly = division(poly, irreduciblePoly)[0]
poly = map(lambda x: x % modulus, poly)
return poly
T.<t> = PolynomialRing(ZZ, 'x')
a = gen()
print Y(a).list() == T(ring_correct(a, ring, modulus)).list()
```

Thanks in advance.