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I computed sum of power residue symbol in N th cyclotomic field:

def new_residue_symbol(a,P,N):
    if a not in P:
        return K(a).residue_symbol(P,N)
    else:
        return 0
N=3
K.<z>=CyclotomicField(N)
p=5
P =(K.ideal(p).factor())[0][0]#P is prime ideal above p
sum(new_residue_symbol(a,P,N) for a in P.residue_field())

This must be $0$ but sage put error typeerror: unable to convert 0 to Cyclotomic Field of order 3 and degree 2. How can I fix this? (When you change $p=7$, then sage put $0$. This might be lifting problem of representatives of residue field )

If you don’t mind, you can edit your answer with some description. Your answer will be better than mine :).

I reported there, but that’s not a bug.

On power residue symbol I computed sum of power residue symbol in N th cyclotomic field: def new_residue_symbol(a,P,N):

I improved the code, please check.

On computation of Jacobi sum 2 I’m computing Jacobi sum of power residue symbol as follows: def new_residue_symbol(a,P,

On computation of Jacobi sum 2 I’m computing Jacobi sum of power residue symbol as follows: def Jacobi(P,l,j): x=su

On computation of Jacobi sum 2 I’m computing Jacobi sum of power residue symbol as follows: def Jacobi(P,l,j): x=su

On computation of Jacobi sum 2 I’m computing Jacobi sum of power residue symbol as follows: def Jacobi(P,l,j): x=su

Oh my goodness.Thank you.

On computation of Jacobi sum I’m computing Jacobi sum of cubic residue symbol as follows: N=3 x=polygen(ZZ,'x') K.<z

On computation of Jacobi sum I’m computing Jacobi sum of cubic residue symbol as follows: N=3 x=polygen(ZZ,'x') K.<z

I want to compute quotient of integer ring of $\mathbb{Q}(\omega) (\omega^3=1)$ by a prime ideal $(-4-3\omega)$. Especially I want to compute representatives.

N=3
x=polygen(ZZ,'x')
K.<a>=CyclotomicField(N)
O = K.ring_of_integers()
p=-4-3*a
R.<b,c>=QuotientRing(O, K.ideal(p))

What should I do next?

More, I want to compute its cardinality (=13), But

R.cardinality()

made error.

This code no longer works. See this forum.

Thank you. What does R(a) mean? By the way this way seems very ad hoc. Isn’t there any ways to compute representatives o

Thank you again! The problem solved by following code! S=[((3+sqrt(8))^i+(3-sqrt(8))^i)/2 for i in [1..40]] [ZZ(a.expan

I want to do prime factorization of following $a$

a=sqrt(2)+6-sqrt(2)
a.factor()

But sage says simply

6

What should I do?

(I originally wanted to factor $\frac{(3+\sqrt{8})^i +(3-\sqrt{8})^i}{2}$ for i in [1..40])

Thank you again! The problem solved by following code! S=[((3+sqrt(8))^i+(3-sqrt(8))^i)/2 for i in [0..40]] [ZZ(a.expan

How can I obtain representatives of a quotient ring? I want to compute quotient of integer ring of $\mathbb{Q}(\omega) (

Thank you. Then I want to move on original question: a=((3+sqrt(8))^3+(3-sqrt(8))^3)/2 ZZ(a).factor() But then sage p

How to factor an integer apparently including irrational numbers I want to do prime factorization of following $a$ a=sq

How to factor an integer apparently including irrational numbers I want to do prime factorization of following $a$ a=sq

How to factor an integer apparently including irrational numbers I want to do prime factorization of following $a$ a=sq

This code no longer works. See this forum .